text:
\begin{document}
\begin{abstract}
We review recent results concerning closed G$_2$-structures on seven-dimensional manifolds.
In particular, we discuss the construction of examples and some related problems.
\end{abstract}
\maketitle

\section{Introduction}
Let $M$ be a seven-dimensional manifold. A $\mathrm{G}_2$-reduction of its frame bundle, i.e., a $\mathrm{G}_2$-{\em structure}, is characterised by the existence of a nowhere vanishing differential
form $\varphi\in\Omega^3(M)$ with pointwise stabiliser isomorphic to the exceptional Lie group $\mathrm{G}_2$.
Any such 3-form $\varphi$ gives rise to a Riemannian metric $g_\varphi$ and to an orientation on $M,$ and its covariant derivative $\nabla^{g_\varphi}\varphi$ with respect to the Levi Civita connection
of $g_\varphi$ represents the obstruction to having $\mathrm{Hol}(g_\varphi)\subseteq\mathrm{G}_2$.
By \cite{FG}, a $\mathrm{G}_2$-structure $\varphi$ is {\em parallel}, i.e., $\nabla^{g_\varphi}\varphi=0$, if and only if both $d\varphi=0$ and $d*_\varphi\varphi=0$, where $*_\varphi$ is the Hodge operator
associated with $g_\varphi$ and the given orientation. A $\mathrm{G}_2$-structure satisfying the former condition is called {\em closed}, and a $\mathrm{G}_2$-structure satisfying the latter is called {\em co-closed}.

All known and potentially effective methods to obtain parallel $\mathrm{G}_2$-structures on compact 7-manifolds require the existence of a closed $\mathrm{G}_2$-structure, possibly satisfying
some additional properties \cite{Bryant,Joyce,Kov,CHNP,KoLe,JoKa,LW}.
However, currently there are no known results guaranteeing the existence of closed $\mathrm{G}_2$-structures on compact manifolds,
neither it is known whether the existence of this type of $\mathrm{G}_2$-structures imposes any constraint on the properties of the manifold, e.g.~on its topology.

In this survey, we review various known examples of compact and non-compact 7-manifolds admitting closed $\mathrm{G}_2$-structures, and we discuss the techniques used to obtain them.
Moreover, we give a new example satisfying some remarkable properties, and we present some related open problems.

\section{Preliminaries}

A $\mathrm{G}_2$-structure on a seven-dimensional manifold $M$ is a reduction of the structure group of its frame bundle to the exceptional Lie group $\mathrm{G}_2$.
Any such reduction occurs if and only if $M$ is orientable and spin \cite{Gra}, and it is characterised by the existence of a 3-form $\varphi\in\Omega^3(M)$ which can be written at each point $x$ of $M$ as
\begin{equation}\label{G2FormAdapted}
\varphi|_x = e^{127}+e^{347}+e^{567} + e^{135} - e^{146} - e^{236} - e^{245},
\end{equation}
with respect to some basis $\left\{e^1,\ldots,e^7\right\}$ of the cotangent space $T^*_xM$.
Here and henceforth, the notation $e^{ijk\cdots}$ is used as a shorthand for the wedge product of covectors $e^i \wedge e^j \wedge e^k \wedge \cdots$.
Any differential 3-form $\varphi$ defining a $\mathrm{G}_2$-structure is a section of an open subbundle $\Lambda^3_+(T^*M) \subset \Lambda^3(T^*M)$
with typical fibre isomorphic to $\mathrm{GL}(7,\mathbb{R})/\mathrm{G}_2$. Therefore, small perturbations of $\varphi$ still define a $\mathrm{G}_2$-structure.
We let $\Omega^3_+(M)\coloneqq \Gamma(\Lambda^3_+(T^*M))$.

Since $\mathrm{G}_2\subset \mathrm{SO}(7)$, every $\mathrm{G}_2$-structure $\varphi$ gives rise to a Riemannian metric $g_\varphi$ and to an orientation on $M.$
In particular, $g_\varphi$ and the corresponding Riemannian volume form $dV_\varphi$ are related to $\varphi$ as follows
\begin{equation*}
g_\varphi(X,Y)\,dV_\varphi = \frac16\,\iota_X\varphi\wedge\iota_Y\varphi\wedge\varphi,
\end{equation*}
for all $X,Y\in\Gamma(TM)$.
We denote by $\nabla^{g_\varphi}$ the Levi Civita connection of $g_\varphi$, and by $*_\varphi$ the Hodge operator associated with $g_\varphi$ and the given orientation.

Given a $\mathrm{G}_2$-structure $\varphi\in\Omega^3_+(M)$, the obstruction to having $\mathrm{Hol}(g_\varphi)\subseteq\mathrm{G}_2$ is represented by the {\em intrinsic torsion} $T_\varphi$.
The latter is a section of a vector bundle over $M$ with typical fibre $(\mathbb{R}^7)^*\otimes \mathfrak{g}_2^\perp$,
where $\mathfrak{g}_2^\perp$ is the orthogonal complement in $\mathfrak{so}(7)$ of the Lie algebra $\mathfrak{g}_2$ of $\mathrm{G}_2$.
It is well-known that $T_\varphi$ can be identified with the covariant derivative $\nabla^{g_\varphi}\varphi$, so that one has $\mathrm{Hol}(g_\varphi)\subseteq\mathrm{G}_2$ if and only if $\nabla^{g_\varphi}\varphi=0$.
When this happens, the $\mathrm{G}_2$-structure is said to be {\em parallel} or {\em torsion-free}.
A further characterisation for parallel $\mathrm{G}_2$-structures was obtained by Fern\'andez and Gray in \cite{FG}. We recall it in the following.
\begin{theorem}[\cite{FG}]\label{FeGrThm}
A $\mathrm{G}_2$-structure $\varphi$ is parallel if and only if both $d\varphi=0$ and $d*_\varphi\varphi=0$.

\end{theorem}

As proved by Bonan in \cite{Bonan}, the metric induced by a parallel $\mathrm{G}_2$-structure is Ricci-flat. Thus, the construction of compact examples is not an easy task.
For instance, compact homogeneous examples are exhausted by flat tori and, more generally, compact 7-manifolds with a parallel $\mathrm{G}_2$-structure $\varphi$ such that
$\mathrm{Hol}(g_\varphi)=\mathrm{G}_2$ cannot have continuous symmetries.

Indeed, using the Bochner-Weitzenb\"ock technique, it is possible to show that the Lie algebra of the automorphism group $\mathrm{Aut}(M,\varphi)$ of a $\mathrm{G}_2$-structure $\varphi$
\begin{equation*}
\mathfrak{aut}(M,\varphi) \coloneqq \left\{X\in\Gamma(TM)~|~\mathcal{L}_X\varphi=0 \right\} \subseteq \mathfrak{isom}(M,g_\varphi)
\end{equation*}
is abelian with $\mathrm{dim} (\mathfrak{aut}(M,\varphi)) = 0, 1, 3, 7$, whenever $\varphi$ is parallel and $M$ is compact.
The possible dimensions depend on having $\mathrm{Hol}(g_\varphi) = \mathrm{G}_2,~\mathrm{SU}(3),~\mathrm{SU}(2),~\{1\}$, respectively.

Compact examples of Riemannian manifolds with holonomy equal to $\mathrm{G}_2$ were constructed in \cite{Joyce,Kov,CHNP,KoLe,JoKa} using techniques such as orbifolds resolutions,
generalised connected sums, and geometric glueing.
All these manifolds admit a parallel $\mathrm{G}_2$-structure obtained by perturbing a $\mathrm{G}_2$-structure $\varphi$ satisfying the condition $d\varphi=0$ and whose intrinsic torsion is {\em small} in a suitable sense.

The manifold topology plays a role in this construction, too, as the metric induced by a parallel $\mathrm{G}_2$-structure on a compact manifold $M$ has holonomy equal to $\mathrm{G}_2$ if and only if
$\pi_1(M)$ is finite \cite{Joyce}.

\section{Closed $\mathrm{G}_2$-structures}

From Theorem \ref{FeGrThm}, we know that a $\mathrm{G}_2$-structure $\varphi$ is parallel if and only if $d\varphi=0$ and $d*_\varphi\varphi=0$.
A $\mathrm{G}_2$-structure satisfying the former condition is called {\em closed}, and one satisfying the latter is called {\em co-closed}.
As we have previously recalled, closed $\mathrm{G}_2$-structures play a central role in the known constructions of compact 7-manifolds with holonomy $\mathrm{G}_2$.
However, currently there are no general results ensuring the existence of closed $\mathrm{G}_2$-structures on compact manifolds,
neither it is known whether the existence of such type of structure imposes any constraint on the manifold structure, e.g.~on its topology.
On the other hand, Crowley and Nordstr\"om proved that the existence of co-closed $\mathrm{G}_2$-structures on compact 7-manifolds is only a topological matter.
Indeed, they always exist on any compact manifold admitting $\mathrm{G}_2$-structures \cite{CrNo}.

\medskip
By \cite{HL}, a closed $\mathrm{G}_2$-structure $\varphi\in\Omega^3_+(M)$ defines a calibration on $M,$ that is, for every oriented tangent 3-plane $V\subset T_xM$ one has $\varphi|_V \leq \mathrm{vol}_V,$
where $\mathrm{vol}_V$ denotes the volume form of the inner product $g_\varphi|_V$ on $V$.
A three-dimensional oriented submanifold $N\hookrightarrow M$ is said to be {\em associative} if it is calibrated by $\varphi$,
i.e., if $\varphi|_{T_xN} = \mathrm{vol}_{T_xN}$ for all $x\in N$. Recall that every compact associative 3-fold is volume minimising in its homology class (see \cite[Thm.~II.4.2]{HL}).
Similarly, if a $\mathrm{G}_2$-structure $\varphi$ is co-closed, then the 4-form $*_\varphi\varphi$ defines a calibration on $M,$ and one can consider four-dimensional oriented submanifolds of $M$ calibrated by it.
These submanifolds are called {\em co-associative}. It is possible to show that a four-dimensional oriented submanifold $U$ in $M$ is co-associative if and only if $\varphi|_U\equiv0$ (cf.~\cite[Cor.~IV.1.20]{HL}).
Thus, when a $\mathrm{G}_2$-structure $\varphi$ is closed but not co-closed, one can define an oriented 4-fold $U$ of $M$ to be co-associative if $\varphi|_U\equiv0$.
However, $U$ is not necessarily volume minimising in its homology class, as $d*_\varphi\varphi\neq0$.
For more details on associative and co-associative submanifolds, we refer the reader to \cite{HL,JoyceBook}.

\medskip
Let us now focus on a 7-manifold $M$ endowed with a closed $\mathrm{G}_2$-structure $\varphi$.
In this case, the intrinsic torsion $T_\varphi$ can be identified with a unique 2-form $\tau\in\Omega^2_{14}(M)\coloneqq \left\{\alpha\in\Omega^2(M)~|~ \alpha \wedge \varphi = -*_\varphi \alpha  \right\}$ such that
\begin{equation*}
d  *_\varphi \varphi = \tau \wedge \varphi.
\end{equation*}
We shall refer to $\tau$ as the {\em intrinsic torsion form} of $\varphi$. Notice that $\tau$ vanishes identically if and only if $d*_\varphi\varphi=0$, that is, if and only if the closed $\mathrm{G}_2$-structure is parallel.

The Ricci tensor and the scalar curvature of the Riemannian metric $g_{\varphi}$ induced by $\varphi$ can be expressed in terms of the intrinsic torsion form $\tau$ as follows (cf.~\cite{Bryant}):
\begin{eqnarray*}
\mathrm{Ric}  (g_{\varphi})	&=&  \frac 14 |\tau |^2 g_{\varphi} - \frac 14 j_\varphi \left(d \tau - \frac 12 *_\varphi (\tau \wedge \tau)\right), \\
\mathrm{Scal} (g_{\varphi}) 	&=& - \frac 12  | \tau |^2,
\end{eqnarray*}
where the map $j_\varphi:\Omega^3(M)\rightarrow \mathcal{S}^2(M)$ assigns to each 3-form $\gamma$ the symmetric 2-covariant tensor
$j_\varphi (\gamma) (X, Y) = *_\varphi (\iota_X \varphi \wedge \iota_Y \varphi \wedge \gamma).$
In particular, the scalar curvature is always non-positive, and it vanishes identically if and only if $\varphi$ is parallel.
Further properties of $\mathrm{Ric}(g_{\varphi})$ and $\mathrm{Scal}(g_{\varphi})$ when $M$ is compact are summarised in the next theorem.

\begin{theorem}[\cite{Bryant,CI}]\label{CurvClosedCpt}
Let $M$ be a compact 7-manifold endowed with a closed $\mathrm{G}_2$-structure $\varphi$. Then,
\begin{enumerate}[$(1)$]
\item $g_{\varphi}$ is Einstein if and only if $\varphi$ is parallel;
\smallskip
\item \label{ERPint} $ \int_{M} [ \mathrm{Scal} (g_{\varphi})]^2 d V_{\varphi} \leq 3 \int_M   | \mathrm{Ric}(g_{\varphi}) |^2 d V_{\varphi}$.
\end{enumerate}
\end{theorem}

\begin{remark}
Currently, it is still not known whether there exist (even incomplete) closed non-parallel $\mathrm{G}_2$-structures inducing an Einstein metric.
A negative answer when the 7-manifold is a simply connected solvable Lie group endowed with a left-invariant closed $\mathrm{G}_2$-structure was given in \cite{FFM1}.
\end{remark}

In \cite{Bryant}, Bryant proved that the inequality in point \ref{ERPint}.~of Theorem \ref{CurvClosedCpt} reduces to an equality if and only if
\begin{equation*}
d \tau = \frac{ | \tau |^2}{6}   \varphi + \frac{1}{6} *_{\varphi} (\tau \wedge \tau).
\end{equation*}
A closed $\mathrm{G}_2$-structure $\varphi$ satisfying the above equation is called {\em extremally Ricci pinched} ({\em ERP} for short).
Notice that this definition makes sense also in the non-compact setting.

The existence of an ERP $\mathrm{G}_2$-structure on a compact 7-manifold imposes strong constraints on its geometry. In detail:
\begin{proposition}[\cite{Bryant}]\label{ERPprop}
Let $M$ be a compact 7-manifold endowed with an ERP $\mathrm{G}_2$-structure $\varphi$ with intrinsic torsion form $\tau\in\Omega^{2}_{14}(M)$ not identically vanishing.
Then
\begin{enumerate}[$(1)$]
\item $\tau$ has constant (non-zero) norm and $\tau\wedge\tau\wedge\tau=0$;
\item $\tau\wedge\tau$ and $*_\varphi(\tau\wedge\tau)$ are non-zero closed simple forms of constant norm.
Consequently, the tangent bundle of $M$ splits into the orthogonal direct sum of two integrable subbundles $TM=P\oplus Q$, with $P\coloneqq \mathrm{ker}(\tau\wedge\tau)$,
$Q\coloneqq \mathrm{ker}(*_\varphi(\tau\wedge\tau))$.
Moreover, the $P$-leaves are associative submanifolds calibrated by $-|\tau|^{-2}*_\varphi(\tau\wedge\tau)$, and the $Q$-leaves are coassociative submanifolds calibrated by
$-|\tau|^{-2}(\tau\wedge\tau)$;
\item the Ricci tensor of $g_\varphi$ is given by $\mathrm{Ric}(g_\varphi) = \frac{1}{12}\,j_\varphi(*_\varphi(\tau\wedge\tau))= -\frac16\, |\tau|^2\, g_\varphi|_P$.
Hence, it is non-positive with eigenvalues $-\frac16\, |\tau|^2$ of multiplicity three and $0$ of multiplicity four.
\end{enumerate}
\end{proposition}

So far, the only known compact examples were obtained in \cite{Bryant,KaLa}. The example in \cite{Bryant} consists of the quotient $\Gamma\backslash M$, where
$M=\mathrm{SL}(2, {\mathbb C}) \ltimes {\mathbb C}^2 /\mathrm{SU}(2)$ is a non-compact homogeneous space endowed with an invariant ERP $\mathrm{G}_2$-structure,
and $\Gamma \subset \mathrm{Aut}(M, \varphi)$ is a co-compact discrete subgroup. The example in \cite{KaLa} is the compact quotient of a simply connected solvable Lie group
endowed with the left-invariant ERP G$_2$-structure obtained in \cite[Ex.~4.7]{Lauret2}.
Further examples are given in \cite{Ball, FRerp, LN,LN2}.

\section{Examples of manifolds admitting closed $\mathrm{G}_2$-structures}

Since there are no general results ensuring the existence of closed $\mathrm{G}_2$-structures on compact 7-manifolds, one should in principle try to solve the PDEs arising from the equation $d\varphi=0$
for a generic 3-form $\varphi\in\Omega^3_+(M)$ on a given compact manifold $M.$ In some cases, this problem may be reduced to a simpler one.
In this section, we give an overview of various known examples and we discuss some new ones.

\subsection{Closed G$_{\mathbf2}$-structures with symmetry}\label{ClosedG2Sym}
The first examples of compact 7-manifolds admitting closed non-parallel $\mathrm{G}_2$-structures were obtained by Fern\'andez in \cite{Fer1,Fer2}.
These examples are of the form $M=\Gamma\backslash\mathrm{G}$, where $\mathrm{G}$ is a seven-dimensional simply connected nilpotent or solvable Lie group,
$\Gamma\subset\mathrm{G}$ is a co-compact discrete subgroup (lattice), and the closed $\mathrm{G}_2$-structure on $M$ is induced by a left-invariant one on $\mathrm{G}$.
In particular, $(M,\varphi)$ is locally homogeneous. Further examples of this type have been obtained in \cite{CF, Freibert,Lauret1}.
We emphasise that in all these examples the Lie group $\mathrm{G}$, being diffeomorphic to $\mathbb{R}^7$, is not compact.
Moreover, any left-invariant $\mathrm{G}_2$-structure $\varphi$ on $\mathrm{G}$ is completely determined by the 3-form $\varphi|_{1_{\mathrm{G}}}$
on the Lie algebra $\mathfrak{g}\cong T_{1_{\mathrm{G}}}\mathrm{G}$, and the condition $d\varphi=0$ is equivalent to a system of algebraic equations for the coefficients of $\varphi|_{1_{\mathrm{G}}}$.

As this last observation suggests, symmetry considerations might help to reduce the PDEs arising from the condition $d\varphi=0$ to equations of a simpler type.
Therefore, one may ask whether there exist examples of compact 7-manifolds $M$ endowed with a closed non-parallel $\mathrm{G}_2$-structure $\varphi$
that are acted on with low cohomogeneity by the automorphism group $\mathrm{Aut}(M, \varphi)$ or a subgroup thereof.
In \cite{CS}, Cleyton and Swann considered 7-manifolds endowed with a $\mathrm{G}_2$-structure and acted on with cohomogeneity one by a simple group of automorphisms,
showing in particular that if $\varphi$ is closed then only complete non-compact examples occur.
Moreover, the existence of compact homogeneous 7-manifolds endowed with an invariant closed non-parallel $\mathrm{G}_2$-structure, i.e., having a transitive group of automorphisms
$\mathrm{G}\subset \mathrm{Aut}(M, \varphi)$, has been open for some time (see \cite[Question 3.1]{Lauret2}).

In \cite{PR}, Podest\`a and the second named author of the present survey studied the properties of $\mathrm{Aut}(M, \varphi)$ when $M$ is compact and the $\mathrm{G}_2$-structure $\varphi$ is closed and non-parallel,
obtaining the following result.

\begin{theorem}[\cite{PR}]\label{ThmAut}
Let $M$ be a compact 7-manifold endowed with a closed non-parallel $\mathrm{G}_2$-structure $\varphi$.
Then, $ {\mathfrak {aut}} (M, \varphi)$ is abelian with dimension bounded by $\mathrm{min}\{6,b_2(M)\}$.
\end{theorem}

The above result immediately implies that there are no compact homogeneous 7-manifolds endowed with an invariant closed non-parallel $\mathrm{G}_2$-structure, and that the only cohomogeneity one examples
occur on manifolds diffeomorphic to the 7-torus.

Notice that the examples of seven-dimensional simply connected Lie groups $\mathrm{G}$ endowed with a left-invariant closed $\mathrm{G}_2$-structure $\varphi$ mentioned earlier (\cite{Fer1,Fer2,CF, Freibert,Lauret1})
correspond to the case when $M$ is not compact and $\mathrm{G}\subset \mathrm{Aut}(M,\varphi)$ acts simply transitively on it.
Recently, the complete classification of non-compact 7-manifolds admitting closed non-parallel $\mathrm{G}_2$-structures and acted on transitively by a reductive group of automorphisms has been obtained in \cite{PR4}.

We remark that Theorem \ref{ThmAut} gives some insights on a currently open problem concerning the existence of a closed $\mathrm{G}_2$-structure on the 7-sphere $\mathbb{S}^7$,
as it implies that $\mathrm{Aut}(\mathbb{S}^7,\varphi)$ must be finite for any hypothetical closed $\mathrm{G}_2$-structure $\varphi\in\Omega^3_+(\mathbb{S}^7)$.

\subsection{A compact example obtained via the resolution of an orbifold}

Besides symmetry considerations, other techniques may be used to construct examples of compact 7-manifolds admitting closed $\mathrm{G}_2$-structures.
For instance, all known examples of compact manifolds admitting parallel $\mathrm{G}_2$-structures are endowed with a closed one.
The techniques used to obtain these examples include orbifolds resolutions \cite{Joyce}, the twisted connected sum construction \cite{Kov,CHNP,KoLe},
and analysis on families of Eguchi-Hanson spaces \cite{JoKa}.

Recently, Fern\'andez, Kovalev, Mu\~noz and the first named author of the present survey obtained a new example of a compact 7-manifold $M$ with $b_1(M)=1$ and admitting closed $\mathrm{G}_2$-structures
but no parallel $\mathrm{G}_2$-structures  \cite{FFKM}.
This example is constructed resolving the singularities of a certain orbifold, and the techniques involved in this construction are motivated by those used by Joyce to obtain compact 7-manifolds with holonomy
$\mathrm{G}_2$ via the resolution of orbifolds of the form ${\mathbb T}^7/\mathrm{F}$, where $\mathrm{F} \subset \mathrm{G}_2 $ is a suitable finite subgroup \cite{Joyce}.

The idea of \cite{FFKM} consists in starting with a compact nilmanifold $M =  \Gamma \backslash \mathrm{N}$ endowed with an invariant closed $\mathrm{G}_2$-structure in place of the 7-torus
$\mathbb{T}^7 \cong \mathbb{R}^7/\mathbb{Z}^7$ endowed with the flat $\mathrm{G}_2$-structure induced by the canonical one on $\mathbb{R}^7$.
Here, $\mathrm{N}$ is the simply-connected nilpotent Lie group corresponding to the 3-step nilpotent Lie algebra $\mathfrak{n}=\langle e_1,\ldots,e_7\rangle$ with the following structure equations
\begin{equation*}
[e_1, e_2 ] = - e_4,~[e_1, e_3 ] = - e_5,~[e_1, e_4 ] = - e_6,~[e_1, e_5 ] = - e_7,
\end{equation*}
$\Gamma \cong 2 \mathbb Z \times \mathbb Z^6$ is a lattice in $\mathrm{N}$, and the invariant closed $\mathrm{G}_2$-structure on $M$ is induced by a left-invariant one on $\mathrm{N}$.
One then considers the action of the finite group $\mathrm{F} = \mathbb{Z}_2$ on $\mathrm{N}$ generated by
\begin{equation*}
\rho:\mathrm{N}\rightarrow\mathrm{N},\quad  (x_1, \ldots, x_7) \mapsto (-x_1, - x_2, x_3, x_4, - x_5, - x_6, x_7).
\end{equation*}
This action satisfies $\rho(a b) = \rho(a) \rho (b)$, for all $a,b \in \mathrm{N},$ and $\rho(\Gamma)=\Gamma$.
Thus, $\rho$ induces an action of $\mathbb Z_2$ on $M = \Gamma \backslash \mathrm{N}$, and it is possible to consider the quotient space $\widehat{M} = M / \mathbb{Z}_2$.
The following result can then be proved.

\begin{theorem}[\cite{FFKM}]  \
\begin{enumerate}[$\bullet$]
\item $\widehat{M} = M /{\mathbb Z}_2$ is a compact 7-orbifold with first Betti number $b_1 (\widehat{M})=1$ and admitting an orbifold closed $\mathrm{G}_2$-form $\widehat{\varphi}$.
The singular locus $S$ of $\widehat{M}$ is the disjoint union of 16 copies of $\mathbb{T}^3$.
\item There exists a resolution $\pi: (\widetilde{M}, \widetilde{\varphi})  \rightarrow ( \widehat{M}, \widehat{\varphi})$, where $\widetilde{M}$ is a compact smooth manifold, $b_1(\widetilde{M})  = 1$,
and $\widetilde{\varphi} = \pi^* \widehat{\varphi}$ in the complement of a neighborhood of the exceptional locus $\pi^{-1} (S)$.
Moreover,  $\pi_1(\widetilde{M}) = \mathbb{Z}$, $\widetilde{M}$ is formal and it does not admit any parallel $\mathrm{G}_2$-structure.
\end{enumerate}
\end{theorem}

\subsection{A comparison with Lie algebras admitting symplectic structures}

As we recalled in Section \ref{ClosedG2Sym}, there are many (non-compact) examples of simply connected seven-dimensional Lie groups admitting left-invariant closed $\mathrm{G}_2$-structures,
and for any such example the properties of the $\mathrm{G}_2$-structure may be studied at the Lie algebra level.
This leads one to focusing on seven-dimensional Lie algebras $\mathfrak{g}$ endowed with a $\mathrm{G}_2$-structure $\varphi\in\Lambda^3_+(\mathfrak{g}^*)$
which is closed with respect to the Chevalley-Eilenberg differential $d$ of $\mathfrak{g}$.
In this setting, there are some problems motivated by symplectic geometry that it is worth investigating.

It is known that symplectic structures and closed $\mathrm{G}_2$-structures share similar properties.
Indeed, they are both defined by closed differential forms satisfying the same non-degeneracy condition, namely at each point of the manifold their orbit under the action of the general linear group is open (see \cite{Hit}).
Moreover, the existence of a symplectic structure on an even-dimensional Lie algebra imposes some constraints on it.
For instance, a unimodular Lie algebra admitting a symplectic structure must be solvable \cite{Chu, LM}, and any unimodular Lie algebra cannot admit symplectic structures defined by an exact 2-form \cite{DiMa}.
Thus, it is natural to investigate whether similar results hold for seven-dimensional unimodular Lie algebras admitting closed $\mathrm{G}_2$-structures, too.
In our works \cite{FFR,FRcomp}, we obtained a negative answer to these problems.

In \cite{FRcomp}, we studied the existence of closed $\mathrm{G}_2$-structures on non-solvable unimodular Lie algebras, showing that examples actually occur and obtaining a complete classification of
the possible Lie algebras up to isomorphism.

\begin{theorem}[\cite{FRcomp}]
Let $\mathfrak{g}$ be a seven-dimensional non-solvable unimodular Lie algebra admitting closed $\mathrm{G}_2$-structures, and denote by
$\mathfrak{g} = \mathfrak{s} \ltimes \mathfrak{r}$ its Levi decomposition, where $\mathfrak{r}$ is the radical of $\mathfrak{g}$ and  $\mathfrak{s}$ is a semisimple subalgebra.
Then, $\mathfrak{s} \cong {\mathfrak {sl}} (2, \mathbb{R})$ and
\begin{enumerate}[$\bullet$]
\item if the Levi decomposition is trivial, namely $\mathfrak{g} = \mathfrak{s}  \oplus \mathfrak{r}$, then $\mathfrak{r}$ is centerless,
and there exists a basis $\{e^1,\ldots,e^7\}$ of $\mathfrak{g}^*$ for which its structure equations $(de^1,\ldots,de^7)$ are in the following list
\begin{equation*}
\renewcommand\arraystretch{1.3}
\begin{array}{l}
(-e^{23}, -2 e^{12}, 2 e^{13}, 0, - e^{45}, \frac 12 e^{46},  - e^{47}, \frac12 e^{47});\\
(-e^{23}, -2 e^{12}, 2 e^{13}, 0, - e^{45}, - \mu e^{46}, (1 + \mu) e^{47}),~- 1 < \mu \leq \frac 12;\\
(-e^{23}, -2 e^{12}, 2 e^{13}, 0, - \mu e^{45}, \frac {\mu}{2}  e^{46} - e^{47}, e^{46} + \frac{\mu}{2} e^{47}),~\mu > 0.
\end{array}
\end{equation*}
\item  if the Levi decomposition is not trivial, then $\mathfrak{r} \cong \mathbb{R} \ltimes \mathbb{R}^3$, and there is a basis $\{e^1,\ldots,e^7\}$ of $\mathfrak{g}^*$ for which its structure equations are the following
\begin{equation*}
(-e^{23}, -2 e^{12}, 2 e^{13}, -e^{14} - e^{25}-  e^{47}, e^{15} - e^{34} - e^{57}, 2 e^{67}, 0).
\end{equation*}
\end{enumerate}
\end{theorem}

In our recent joint work with M.~Fern\'andez \cite{FFR}, we focused on closed $\mathrm{G}_2$-structures defined by an exact 3-form $\varphi$ on a unimodular Lie algebra $\mathfrak{g}$,
showing that there exist examples when the third Betti number $b_3(\mathfrak{g})$ of $\mathfrak{g}$ vanishes and when $\mathfrak{g}$ is (2,3)-trivial, i.e., when $b_2(\mathfrak{g})=0=b_3(\mathfrak{g})$.

\subsection{Products, warped products, and Lie algebras extensions}\label{PWPRk1}

Further examples not mentioned so far can be obtained on products of suitable manifolds.
For instance, it is possible to define closed $\mathrm{G}_2$-structures on the product of a 4-manifold endowed with a hypersymplectic structure and the 3-torus $\mathbb{T}^3$ (see e.g.~\cite{FY}),
and on the product of a 6-manifold endowed with a suitable SU(3)-structure and the circle $\mathbb{S}^1$.

Recall that an SU(3)-structure on a 6-manifold $N$ is the data of an almost Hermitian structure $(g,J)$, with corresponding fundamental 2-form $\omega\coloneqq g(J\cdot,\cdot)$, and a complex (3,0)-form
$\Psi  = \psi + i \widehat{\psi}$ satisfying the normalisation condition $3 \psi \wedge \widehat{\psi} = 2\omega^3$.
Moreover, the whole structure is completely determined by the real 2-form $\omega$ and the real 3-form $\psi$, provided that they satisfy suitable conditions (see \cite{Hit} for details).
Given a positive function $f\in\mathcal{C}^\infty(N)$, it is possible to endow the product $N\times \mathbb{S}^1$ with the 3-form
\begin{equation}\label{G2warped}
\varphi = f\omega \wedge ds +\psi,
\end{equation}
which is easily seen to define a $\mathrm{G}_2$-structure with associated Riemannian metric $g_\varphi = g +f^2ds^2,$ where $s$ denotes the coordinate on the circle.
Notice that the manifold $N\times \mathbb{S}^1$ endowed with this metric is none other than the warped product of $N$ and $\mathbb{S}^1$ with warping function $f$ and fibre $\mathbb{S}^1$.
The $\mathrm{G}_2$-structure $\varphi$ on $N\times \mathbb{S}^1$ is closed if and only if the SU(3)-structure on $N$ satisfies
\begin{equation}\label{SU3wpd}
d\omega = -d\log(f) \wedge \omega,\quad d\psi =0.
\end{equation}

Thus, the problem of finding a closed $\mathrm{G}_2$-structure of the form \eqref{G2warped} on $N\times \mathbb{S}^1$ boils down to the construction of a 6-manifold endowed with an
SU(3)-structure satisfying \eqref{SU3wpd}. An example on $N=\mathbb{T}^6$ can be found in \cite{FRpisa}.

When the warping function $f$ is constant, the manifold reduces to a Riemannian product, and it is always possible to assume that $f\equiv1$ up to scaling the coordinate $s$.
In this case, the $\mathrm{G}_2$-structure $\varphi = \omega \wedge ds + \psi$ is closed if and only if both $d\omega=0$ and $d\psi=0$.
An SU(3)-structure satisfying these conditions is called {\em symplectic half-flat}, and its intrinsic torsion can be identified with the unique real 2-form $w_2$ of type (1,1) such that $w_2\wedge\omega^2=0$ and
$d\widehat{\psi} = w_2 \wedge\omega.$ A simple computation shows that the intrinsic torsion form of  the closed $\mathrm{G}_2$-structure $\varphi = \omega \wedge ds + \psi$ coincides with $w_2$.
We refer the reader to \cite{FMOU,FRpisa,PR2,PR3} and the references therein for more details and examples.

The last example of closed $\mathrm{G}_2$-structures on $N\times\mathbb{S}^1$ that we would like to mention is obtained as follows.
Let us consider a {\em coupled} SU(3)-structure on $N$, i.e., an SU(3)-structure satisfying the condition $d\omega = c\psi$ for some non-zero real number $c$.
The 3-form $\psi$ is clearly closed, while the 3-form $\widehat{\psi}$ satisfies
\begin{equation}\label{cpdhatpsi}
d\widehat{\psi} = -\frac23c\,\omega^2 +w_2\wedge\omega,
\end{equation}
for a unique real 2-form $w_2$ of type (1,1) and satisfying $w_2\wedge\omega^2=0$. The pair $(c,w_2)$ completely determines the intrinsic torsion of the coupled SU(3)-structure $(\omega,\psi)$.
Now, the $\mathrm{G}_2$-structure $\varphi = \omega \wedge ds+\psi$ on $N\times\mathbb{S}^1$ considered before is no longer closed, but there exists a global conformal change making it into a closed one, namely
\begin{equation*}
\widetilde{\varphi}  = e^{cs}\left(\omega\wedge ds+\psi\right).
\end{equation*}
The Riemannian metric induced by $\widetilde{\varphi}$ is given by $g_{\widetilde{\varphi}} = e^{\frac23 cs} g_\varphi$,
and its intrinsic torsion form is $\widetilde{\tau} = e^{\frac{cs}{3}}w_2$.
For more details and examples regarding coupled SU(3)-structures and $\mathrm{G}_2$-structures that are (locally) conformal to closed ones, the reader may refer to \cite{FFR0,FRmz}.

\medskip

As for Lie algebras, similar constructions can be made considering rank-one extensions of six-dimensional Lie algebras endowed with an SU(3)-structure.

We recall that the {\em rank-one extension} of an $n$-dimensional Lie algebra $\mathfrak{h}$ induced by a derivation $D\in\mathrm{Der}(\mathfrak{h})$ is the $(n+1)$-dimensional Lie algebra given by the vector space
$\mathfrak{h}\oplus\mathbb{R}$ endowed with the Lie bracket
\[
\left[(X,a),(Y,b)\right]\coloneqq \left([X,Y]_\mathfrak{h} + a\,D(Y)-b\,D(X),0\right),
\]
where $(X,a),~(Y,b)\in\mathfrak{h}\oplus\mathbb{R}$, and $[\cdot,\cdot]_\mathfrak{h}$ is the Lie bracket on $\mathfrak{h}$. We shall denote this Lie algebra by $\mathfrak{h} {\rtimes_D} \mathbb{R}$.
Moreover, we let $\xi\coloneqq(0,1)$, and we denote by $\eta$ the 1-form on $\mathfrak{h} {\rtimes_D} \mathbb{R}$ such that $\eta(\xi)=1$ and $\mathfrak{h} = \mathrm{ker}(\eta)$.

Let $d$ and $\hat{d}$ denote the Chevalley-Eilenberg differential on $\mathfrak{h} {\rtimes_D} \mathbb{R}$ and on $\mathfrak{h}$, respectively.
Then, for every $k$-form $\gamma\in\Lambda^k(\mathfrak{h}^*)$ the following identity holds:
\begin{equation*}
d\gamma = \hat{d} \gamma +(-1)^{k+1}D^*\gamma\wedge\eta,
\end{equation*}
where the natural action of an endomorphism $A\in\mathrm{End}(\mathfrak{h})$ on $\Lambda^k(\mathfrak{h}^*)$ is given by
\[
A^*\gamma(X_1, \ldots, X_k) = \gamma(AX_1,\ldots,X_k) +\cdots+ \gamma(X_1,\ldots,AX_k),
\]
for all $X_1,\ldots,X_k\in\mathfrak{h}$. If $D=0\in\mathrm{Der}(\mathfrak{h})$ is the trivial derivation, then the rank-one extension of $\mathfrak{h}$ reduces to the product Lie algebra $\mathfrak{h}\oplus\mathbb{R}$.

Assume now that $\mathfrak{h}$ is six-dimensional and that it is endowed with an SU(3)-structure $(\omega,\psi)$. Then, the rank-one extension $\mathfrak{h} {\rtimes_D} \mathbb{R}$ is always endowed
with a $\mathrm{G}_2$-structure defined by the 3-form $\varphi = \omega\wedge \eta +\psi$.
Clearly, the condition $d\varphi=0$ constrains the possible types of SU(3)-structures and the properties of the derivation $D$ that one can consider. Indeed, it holds if and only if
\begin{equation}\label{closedrk1ext}
\hat{d}\omega = -D^*\psi, \quad \hat{d}\psi=0.
\end{equation}
In the next proposition, whose proof immediately follows from \eqref{closedrk1ext}, we recall some relevant cases related to the SU(3)-structures mentioned above.
\begin{proposition}\label{closedG2rk1}
Let $\mathfrak{h}$ be a six-dimensional Lie algebra endowed with an $\mathrm{SU}(3)$-structure $(\omega,\psi)$, and consider the rank-one extension $\mathfrak{h} {\rtimes_D} \mathbb{R}$ endowed with
the $\mathrm{G}_2$-structure $\varphi = \omega\wedge \eta +\psi$. Then, $\varphi$ is closed in the following cases
\begin{enumerate}[$\bullet$]
\item the $\mathrm{SU}(3)$-structure is symplectic half-flat, i.e., $\hat{d}\omega=0$, $\hat{d}\psi=0$, and either $D=0$ or $D^*\psi=0$;
\item the $\mathrm{SU}(3)$-structure is coupled with $\hat{d}\omega=c\psi$, and $D^*\psi = -c\psi.$
\end{enumerate}
\end{proposition}

The classification of six-dimensional solvable Lie algebras admitting symplectic half-flat SU(3)-structures was given in \cite{FMOU}.
Examples of closed $\mathrm{G}_2$-structures on rank-one extensions of these Lie algebras are discussed in \cite{Man}.
\medskip

In \cite{FRmz}, we proved that a six-dimensional nilpotent Lie algebra admitting coupled SU(3)-structures is isomorphic to one of the following
\begin{equation*}
\mathfrak{n}_1 = (0,0,0,e^{13},e^{14}+e^{23},e^{13}-e^{15}-e^{24}), \quad \mathfrak{n}_2 =  (0,0,0,0,e^{14}+e^{23},e^{13}-e^{24}).
\end{equation*}
In both cases, there exists a coupled SU(3)-structure with {\em adapted} basis $\{e^1,\ldots,e^6\}$, namely
\begin{equation}\label{coupledN}
\omega = e^{12} + e^{34} + e^{56},\quad \Psi = (e^1+ie^2) \wedge (e^3+ie^4) \wedge (e^5+ie^6),
\end{equation}
and it is easy to check that $\hat{d}\omega = -\psi$, i.e., $c=-1$.
Moreover, for the coupled SU(3)-structure on $\mathfrak{n}_2$, the unique 2-form $w_2$ fulfilling \eqref{cpdhatpsi} is given by $w_2=\frac43 e^{12} +\frac43 e^{34} -\frac83 e^{56}$ and its
differential $\hat{d}w_2$ is proportional to $\psi$.

\begin{remark}\label{remW2PropPsi}
More generally, if $(\omega,\psi)$ is a coupled SU(3)-structure on a 6-manifold with $dw_2$ proportional to $\psi$, then one has $dw_2 = \frac{|w_2|^2}{4}\psi$ with $|w_2|$ constant (see e.g.~\cite{FRsym}).
Furthermore, on the nilpotent Lie algebra $\mathfrak{n}_1$ there are no coupled SU(3)-structures satisfying this condition (cf.~\cite[Prop.~12]{FRsym}).
\end{remark}

The nilpotent Lie algebra $\mathfrak{n}_2$ endowed with the coupled SU(3)-structure $(\omega,\psi)$ given in \eqref{coupledN} admits rank-one extensions for which
the $\mathrm{G}_2$-structure $\varphi=\omega\wedge\eta+\psi$ is closed.
A one-parameter family of examples was given by Lauret in \cite[Ex.~4.10]{Lauret2}, and it consists of the extension $\mathfrak{g}_a \coloneqq \mathfrak{n}_2\rtimes_{D_a}\mathbb{R}$ with respect to the derivation
\begin{equation*}
D_a = \mathrm{diag}\left(a,a,\frac12-a,\frac12-a,\frac12,\frac12\right) \in \mathrm{Der}(\mathfrak{n}_2),\quad  a \geq \frac14.
\end{equation*}
Notice that $D_a$ fulfills the condition $D_a^*\psi=\psi$, whence $d\varphi=0$.
The Lie algebra $\mathfrak{g}_a$ is completely solvable, and the bound on the real parameter $a$ arises requiring the simply connected Lie groups $\mathrm{G}_a$ with Lie algebra $\mathfrak{g}_a$ to be
pairwise non-isomorphic for different values of $a$.

The Lie algebra $\mathfrak{n}_1$ endowed with the coupled SU(3)-structure given in \eqref{coupledN} admits derivations satisfying the condition $D^*\psi=\psi$, too.
It is not difficult to prove that the most general one has the following expression with respect to the basis $\{e_1,\ldots,e_6\}$ of $\mathfrak{n}_1$
\begin{equation*}
D_{a,b} =
\begin{pmatrix}
0	&	0	&	0	&	0	&	0	&	0	\\
0	&	0	&	0	&	0	&	0	&	0	\\
a	&	0	&\frac12	&	0	&	0	&	0	\\
0	&	a	&	0	&\frac12	&	0	&	0	\\
b	&	0	&	0	&	0	&\frac12	&	0	\\
0	&	b	&	0	&	0	&	0	&\frac12
\end{pmatrix}, \quad a,b\in\mathbb{R}.
\end{equation*}
Consequently, any Lie algebra belonging to the two-parameter family $\mathfrak{g}_{a,b} \coloneqq \mathfrak{n}_1 \rtimes_{D_{a,b}}\mathbb{R}$ admits a closed $\mathrm{G}_2$-structure
defined by $\varphi=\omega\wedge\eta+\psi$.
\medskip

We now discuss a new example of a solvable non-nilpotent Lie algebra endowed with a coupled SU(3)-structure, and we show that there exist rank-one extensions admitting closed $\mathrm{G}_2$-structures.

\begin{example}\label{newcoupled}
Consider the two parameter family of six-dimensional unimodular solvable Lie algebras $\mathfrak{s}_{a,b}$, $a,b\in\mathbb{R}$,
whose structure equations with respect to a basis $\left\{e^1,\ldots,e^6\right\}$ of $\mathfrak{s}^*_{a,b}$ are the following
\begin{equation*}
\begin{cases}
de^1 = -a\,e^{26},\\
de^2 = a\,e^{16},\\
de^3 = b\,e^{16}+b\,e^{25}+a\,e^{46},\\
de^4 = b\,e^{15}-b\,e^{26}-a\,e^{36},\\
de^5=de^6=0.
\end{cases}
\end{equation*}
\end{example}
Notice that $\mathfrak{s}_{a,b}\cong\mathbb{R}^4\rtimes\mathbb{R}^2,$ where the subalgebra $\mathbb{R}^2$ is spanned by the vectors $e_5$ and $e_6$.
We endow $\mathfrak{s}_{a,b}$ with the SU(3)-structure $(\omega,\psi)$ given by \eqref{coupledN}.

If $a=0$, then the Lie algebra $\mathfrak{s}_{0,b}$ is isomorphic to the nilpotent Lie algebra $\mathfrak{n}_2$ and the SU(3)-structure is the coupled one described before.
If $b=0$, then $\mathfrak{s}_{a,0}$ is solvable non-nilpotent, the SU(3)-structure satisfies $\hat{d}\omega=0$, $\hat{d}\psi = 0 = \hat{d}\widehat{\psi}$, and the associated inner product $g=\sum_{k=1}^6(e^k)^2$ is flat.

From now on, we assume that $a\neq0$ and $b\neq 0$. Under this assumption, the Lie algebra is solvable non-nilpotent and the SU(3)-structure is coupled with
\begin{equation*}
\hat{d}\omega = b\,\psi.
\end{equation*}
Moreover, it has $w_2=-\frac43 b e^{12} +\frac83 b e^{34} -\frac43 b e^{56}$, which is easily seen to satisfy the condition $\hat{d}w_2 = \frac83 b^2 \psi$.

There exists a one-parameter family of derivations $D_k\in\mathrm{Der}(\mathfrak{s}_{a,b})$ for which the 3-form $\varphi=\omega\wedge\eta+\psi$ defines a closed $\mathrm{G}_2$-structure
on the rank-one extension $\mathfrak{g}_{a,b,k}\coloneqq\mathfrak{s}_{a,b}\rtimes_{D_k}\mathbb{R}$. It has the following expression
\begin{equation*}
D_k =
\begin{pmatrix}
-\frac{b}{2}	&	k	&	0	&	0	&	0	&	0	\\
-k			&-\frac{b}{2}	&	0	&	0	&	0	&	0	\\
0			&	0	&-\frac{b}{2}	&	-k	&	0	&	0	\\
0			&	0	&	k	&-\frac{b}{2}	&	0	&	0	\\
0			&	0	&	0	&	0	&0	&	0	\\
0			&	0	&	0	&	0	&	0	&0
\end{pmatrix}, \quad k\in\mathbb{R},
\end{equation*}
and it is easily seen to satisfy the equation $D_k^*\psi = -b\,\psi,$ for any $k\in\mathbb{R}$.

\section{Some remarks on closed $\mathrm{G}_2$-structures induced by coupled SU(3)-structures on Lie algebras}

An interesting property of Lauret's example \cite[Ex.~4.10]{Lauret2}, reviewed in Section \ref{PWPRk1}, is that the left-invariant closed $\mathrm{G}_2$-structure induced by $\varphi=\omega\wedge\eta+\psi$
on the simply connected Lie group $\mathrm{G}_a$ is always a {\em Laplacian soliton}, that is,
its Hodge Laplacian $\Delta_\varphi\varphi = -d*_\varphi d*_\varphi\varphi =d\tau$  satisfies the equation
\begin{equation}\label{LapSoliton}
\Delta_\varphi\varphi = \lambda\varphi +\mathcal{L}_{X}\varphi,
\end{equation}
for $\lambda = \lambda(a)\coloneqq 8a^2-4a-4$, and for a suitable vector field $X$ on $\mathrm{G}_a$.
Moreover, this Laplacian soliton is {\em algebraic} according to \cite[Def.~3.9]{Lauret1}, namely there exists a derivation $B\in\mathrm{Der}(\mathfrak{g}_a)$ such that
the vector field $X$ is induced by the unique one-parameter group of automorphisms $\Phi_t\in\mathrm{Aut}(\mathrm{G}_a)$ satisfying
$\left.d\Phi_t\right|_{1_{\mathrm{G}_a}} = \exp(tB)\in\mathrm{Aut}(\mathfrak{g}_a)$.
Finally, depending on the value of $a$, the Laplacian soliton can be {\em shrinking} ($\lambda(a)<0$), {\em steady} ($\lambda(a)=0$), or {\em expanding} ($\lambda(a)>0$).

\begin{remark}\
\begin{enumerate}[(1)]
\item So far, the shrinking Laplacian solitons on $\mathrm{G}_a$, for $\frac14\leq a <1$, constitute the only known examples of such solitons.
\item The left-invariant steady Laplacian soliton, corresponding to the value $a=1$, is also an ERP $\mathrm{G}_2$-structure.
More generally, Lauret and Nicolini proved that every left-invariant ERP $\mathrm{G}_2$-structure on a simply connected Lie group is a steady Laplacian soliton \cite{LN}.
On the other hand, there exists an example of a left-invariant steady Laplacian soliton that is not ERP \cite{FR19}.
\end{enumerate}
\end{remark}

On a (compact) 7-manifold $M,$ all closed $\mathrm{G}_2$-structures satisfying \eqref{LapSoliton} have a distinguished behaviour when they evolve under the $\mathrm{G}_2$-{\em Laplacian flow}.
This flow, which was introduced by Bryant in \cite{Bryant}, prescribes the evolution of a closed $\mathrm{G}_2$-structure $\varphi\in\Omega^3_+(M)$ as follows
\begin{equation}\label{LapFlow}
\begin{cases}
\frac{\partial}{\partial t}\varphi(t) = \Delta_{\varphi(t)}\varphi(t),\\
d\varphi(t)=0,\\
\varphi(0) = \varphi.
\end{cases}
\end{equation}
The initial datum $\varphi$ is a Laplacian soliton, i.e., it satisfies \eqref{LapSoliton} for some real number $\lambda$ and some vector field $X\in\Gamma(TM)$,
if and only if the solution $\varphi(t)$ of \eqref{LapFlow} starting from it at $t=0$ is {\em self-similar}, that is
\begin{equation*}
\varphi(t) = \left(1+\frac23\lambda t\right)^{\frac32} F_t^*\varphi,
\end{equation*}
where $F_t\in\mathrm{Diff}(M)$ is the one-parameter family of diffeomorphisms generated by the vector field $X(t) = \left(1+\frac23\lambda t\right)^{-1}X$ and satisfying $F_0=\mathrm{Id}_M$.
In particular, $\varphi(t)$ exists on the maximal time interval  $\left(-\infty,-\frac{3}{2\lambda}\right)$ when $\lambda<0$, $(-\infty,+\infty)$ when $\lambda=0$, and $\left(-\frac{3}{2\lambda},+\infty\right)$ when $\lambda>0$.
We refer the reader to \cite{Bryant,LW} for more details on the $\mathrm{G}_2$-Laplacian flow and its solitons.
\medskip

In \cite{Lauret2}, Lauret computed the solution of the $\mathrm{G}_2$-Laplacian flow on $\mathfrak{g}_a=\mathfrak{n}_2\rtimes_{D_a}\mathbb{R}$
starting from the closed $\mathrm{G}_2$-structure $\varphi=\omega\wedge\eta+\psi$.
When $a\neq1$, it is given by
\begin{equation*}
\varphi(t) = {A_t}^{q_1(a)}\,e^{127} + {A_t}^{q_2(a)}\,e^{347} + {A_t}^{q_3(a)}\,\left(e^{567} + e^{135} - e^{146} - e^{236} - e^{245} \right),
\end{equation*}
where $e^7\coloneqq \eta$, $A_t = \frac23\,\lambda(a)\,t+1$, and
\begin{equation*}
q_1(a) = \frac{3a}{2(2a+1)},\quad q_2(a) = \frac{3(2a-1)}{8(a-1)},\quad q_3(a) = \frac{9}{8(2a+1)(a-1)}.
\end{equation*}
The maximal interval of existence of $\varphi(t)$ is $\left(-\infty,-\frac{3}{2\lambda(a)}\right)$, when $\frac14\leq a <1$, and $\left(-\frac{3}{2\lambda(a)},+\infty\right)$, when $a >1$.
\medskip

An examination of the solution $\varphi(t)$ allows us to notice the following remarkable property.
\begin{proposition}\label{remflowcpd}
Consider the seven-dimensional Lie algebra $\mathfrak{g}_a=\mathfrak{n}_2\rtimes_{D_a}\mathbb{R}$, for $a\geq\frac14$ with $a\neq1$, and let $\varphi=\omega\wedge \eta+\psi$ be the closed
$\mathrm{G}_2$-structure on it induced by the coupled $\mathrm{SU}(3)$-structure $(\omega,\psi)$ on $\mathfrak{n}_2$ given in \eqref{coupledN}.
Then, the solution of the $\mathrm{G}_2$-Laplacian flow starting from $\varphi$ at $t=0$ can be written as follows
\begin{equation*}
\varphi(t) = \omega(t) \wedge f(t)\,\eta + \psi(t),
\end{equation*}
where $f(t)=(A_t)^{\frac12}$ and the pair
\begin{eqnarray*}
\omega(t) 	&=& {A_t}^{q_1(a)-\frac12}\,e^{127} + {A_t}^{q_2(a)-\frac12}\,e^{347} + {A_t}^{q_3(a)-\frac12}\,e^{567},\\
\psi(t)	&=& {A_t}^{q_3(a)} \left(e^{135} - e^{146} - e^{236} - e^{245}\right) = {A_t}^{q_3(a)}\,\psi,
\end{eqnarray*}
defines a family of coupled $\mathrm{SU}(3)$-structures on $\mathfrak{n}_2$ satisfying
\begin{equation*}
\hat{d}\omega(t) = c(t)\,\psi(t),\quad c(t) = -(A_t)^{-\frac12},
\end{equation*}
and having $\hat{d}w_2(t)$ proportional to $\psi(t)$.
Finally, $\varphi(t)$ induces the metric $g_\varphi(t) = g(t) + [f(t)]^2 \eta^2$,
where $g(t)$ denotes the metric associated with the coupled $\mathrm{SU}(3)$-structure $(\omega(t),\psi(t))$.
\end{proposition}

Let $\mathfrak{h}$ denote a six-dimensional Lie algebra endowed with a coupled SU(3)-structure $(\omega,\psi)$ satisfying $\hat{d}\omega=c\psi$, and assume that there exists a derivation
$D\in\mathrm{Der}(\mathfrak{h})$ such that $D^*\psi=-c\psi$.
The observations given in Proposition \ref{remflowcpd} lead to the question whether the $\mathrm{G}_2$-Laplacian flow on $\mathfrak{h}\rtimes_D\mathbb{R}$ starting from
$\varphi=\omega\wedge\eta+\psi$ can be reinterpreted as a flow on $\mathfrak{h}$ evolving the coupled SU(3)-structure $(\omega,\psi)$.
The examination of the flow on the Lie algebras $\mathfrak{g}_{a,b}$ and $\mathfrak{g}_{a,b,k}$ defined in Section \ref{PWPRk1} may give some insights on this problem.
For the sake of convenience, henceforth we let $e^7\coloneqq \eta$ and $e_7\coloneqq\xi$, so that the closed $\mathrm{G}_2$-structure $\varphi$ on these Lie algebras can be written as in \eqref{G2FormAdapted}
with respect to the basis $\left\{e^1,\ldots,e^6,e^7\right\}$ we are considering.
\medskip

Let us begin with the Lie algebra $\mathfrak{g}_{a,b,k}$.
Notice that the coupled SU(3)-structure $(\omega,\psi)$ on $\mathfrak{s}_{a,b}$ satisfies the additional property $\hat{d}w_2\propto \psi$, which is also satisfied
by the coupled SU(3)-structure considered on $\mathfrak{n}_2$ and is never satisfied by any coupled SU(3)-structure on $\mathfrak{n}_1$ (cf.~Remark \ref{remW2PropPsi}).

We consider the following Ansatz for the solution
\begin{equation}\label{Ansatz}
\varphi(t) = C_1e^{127}+C_2e^{347}+C_3e^{567} +C_4e^{135} -C_5e^{146} -C_6e^{236} -C_7e^{245},
\end{equation}
where $C_i = C_i(t)$, $i=1,\ldots,7$, are unknown positive functions depending only on $t$ and satisfying $C_i(0)=1$.
Then, $\varphi(t)$ still defines a $\mathrm{G}_2$-structure, and it is closed if and only if $C_7 = C_6 = C_5 = C_4 =C_2$.
Now, some straightforward computations show that the intrinsic torsion form $\tau(t)$ of $\varphi(t)$ has the following expression
\begin{equation*}
\tau(t) = -b\left(\frac{C_1{C_2}^2}{{C_3}^2}\right)^{\frac13} e^{12} +3b \left(\frac{{C_2}^5}{{C_1}^2{C_3}^2}\right)^{\frac13} e^{34} -2b \left(\frac{{C_2}^2{C_3}}{{C_1}^2} \right)^{\frac13}e^{56}.
\end{equation*}
Consequently, recalling that $\Delta_{\varphi(t)}\varphi(t) = d\tau(t)$, we have
\begin{equation*}
\Delta_{\varphi(t)}\varphi(t) = -b^2\left(\frac{C_1{C_2}^2}{{C_3}^2}\right)^{\frac13} e^{127} + 3b^2\left(\frac{{C_2}^5}{{C_1}^2{C_3}^2}\right)^{\frac13} \left(e^{347} +e^{135} - e^{146} - e^{236} - e^{245} \right),
\end{equation*}
whence we see that the Laplacian flow equation \eqref{LapFlow} is equivalent to the following system of ODEs for the functions $C_1, C_2, C_3$:
\begin{equation*}
\begin{cases}
\frac{d}{dt} C_1 = -b^2\left(\frac{C_1{C_2}^2}{{C_3}^2}\right)^{\frac13},\\
\frac{d}{dt} C_2 =  3b^2\left(\frac{{C_2}^5}{{C_1}^2{C_3}^2}\right)^{\frac13}, \\
\frac{d}{dt} C_3 = 0.
\end{cases}
\end{equation*}
Combining these ODEs with the initial conditions $C_1(0)=1=C_2(0)=C_3(0)$, we immediately get $C_3(t)=1$ and $C_1 = {C_2}^{-\frac13}$, with $C_2$ solving the Cauchy problem
\begin{equation*}
\begin{cases}
\frac{d}{dt} C_2 = 3b^2{C_2}^{\frac{17}{9}},\\
C_2(0)=1.
\end{cases}
\end{equation*}
Thus, we have $C_2(t) = \left(-\frac83 b^2 t +1\right)^{-\frac98}$, and we see that the solution of the $\mathrm{G}_2$-Laplacian flow starting from $\varphi$ is defined in the interval $\left(-\infty,\frac{3}{8b^2}\right)$.
Moreover, it can be written as
\begin{equation*}
\varphi(t) = \omega(t) \wedge f(t)\,\eta + \psi(t),
\end{equation*}
where $f(t)=  \left(-\frac83 b^2 t +1\right)^{\frac12}$, and $(\omega(t),\psi(t))$ is a family of coupled SU(3)-structures on $\mathfrak{s}_{a,b}$ satisfying $\hat{d}\omega(t)=c(t)\psi(t)$
with $c(t) = b  \left(-\frac83 b^2 t +1\right)^{-\frac12}$ and having $\hat{d}w_2(t)$ proportional to $\psi(t)$.
\medskip

Comparing this discussion with Proposition \ref{remflowcpd}, we see that the solution we have just found shares similar properties with the solution of the Laplacian flow obtained by Lauret on
$\mathfrak{g}_a$, for $\frac14\leq a<1$.
However, we claim that the closed $\mathrm{G}_2$-structure we are considering is never an algebraic soliton.
The proof goes as follows. From \cite{Lauret1}, we know that if $\varphi$ gives rise to a left-invariant algebraic soliton on the simply connected Lie group with Lie algebra $\mathfrak{g}_{a,b,k}$,
then there exist a derivation $B\in\mathrm{Der}(\mathfrak{g}_{a,b,k})$ and a real number $\lambda$ such that
\begin{equation*}
\Delta_\varphi\varphi = d\tau = \lambda \varphi +B^*\varphi.
\end{equation*}
Since $a\neq0$ and $b\neq0$, the space $\mathrm{Der}(\mathfrak{g}_{a,b,k})$ is eight-dimensional,
and we can write a generic derivation with respect to a basis $\{B_1,\ldots,B_8\}$ of $\mathrm{Der}(\mathfrak{g}_{a,b,k})$ as $B = \sum_{i=1}^8 A^iB_i$, for some real parameters $A_i$.
We can then compute the 3-form $B^*\varphi$, and see that the above equation reduces to a system of polynomial equations in the unknowns $A_i$ and $\lambda$.
It turns out that this system has no solutions under the assumption $b\neq0$.
\medskip

On the other hand, obtaining the explicit solution of the $\mathrm{G}_2$-Laplacian flow on $\mathfrak{g}_{a,b}$ satisfying $\varphi(0)=\varphi$ seems challenging.
For instance, the Ansatz \eqref{Ansatz} is not helpful in this case, as the flow equation in \eqref{LapFlow} implies that some of the functions $C_i(t)$ must be identically vanishing, a contradiction.
Notice that in this example the coupled SU(3)-structure does not satisfy the condition $\hat{d}w_2\propto\psi$.
\medskip

We now collect some observations and open problems motivated by the results presented in this last section.
\begin{enumerate}[(1)]
\item Is it possible to determine the explicit expression of the solution of the $\mathrm{G}_2$-Laplacian flow on $\mathfrak{g}_{a,b}$ starting from the closed $\mathrm{G}_2$-structure $\varphi$?
\item Albeit the Lie algebras $\mathfrak{g}_a$, $\frac14\leq a <1$, and $\mathfrak{g}_{a,b,k}$, $a\neq0,~b\neq0$, are not isomorphic, they admit coupled SU(3)-structures sharing similar properties.
Are the corresponding simply connected Lie groups endowed with the left-invariant coupled SU(3)-structures equivalent in a suitable sense?
\item Consider a six-dimensional Lie algebra $\mathfrak{h}$ endowed with a coupled SU(3)-structure $(\omega,\psi)$ with $\hat{d}\omega=c\psi$ and admitting a derivation satisfying $D^*\psi=-c\psi$.
Is it possible to reinterpret the $\mathrm{G}_2$-Laplacian flow on $\mathfrak{g} = \mathfrak{h}\rtimes_D\mathbb{R}$ evolving the closed $\mathrm{G}_2$-structure $\varphi=\omega\wedge\eta+\psi$
as a flow evolving the coupled SU(3)-structure on $\mathfrak{h}$? Does the condition $\hat{d}w_2\propto\psi$ play some role in this problem?
\item Is it possible to classify all six-dimensional solvable Lie algebras admitting a coupled SU(3)-structure up to isomorphism, at least when the condition $\hat{d}w_2\propto\psi$ holds?
\end{enumerate}

\bigskip

\noindent
{\bf Acknowledgements.} The authors were supported by G.N.S.A.G.A.~of I.N.d.A.M.~and by  the project P.R.I.N.~2017 ``Real and Complex Manifolds: Topology, Geometry and Holomorphic Dynamics''.
A.~F.~would like to thank the organisers of the Abel Symposium, J. ~Figueroa-O'Farrill,  S.~Hervik,  B.~Kruglikov, I.~Markina, J.~Slov\'ak, D.~The, and B.~\O rsted,
for the wonderful conference and their hospitality.


TITLE: Bounds of integration
QUESTION [1 upvotes]: A little bit rusty on the topic of surface integrals and perhaps some basic calculus after a while. A little help would be appreciated
Find the volume of the solid under the surface $$z=3x^{2}+y^{2}
$$ and above the region bounded by $$y=x$$ and $$y^{2}-y=x.$$
The main problem I'm facing is with the parabola. I've found the parabola to cut the y-intercept at $y=0$ and $y=1$. However, this does not square with the solutions I am looking at.
Any help is appreciated.

REPLY [1 votes]: It might be easier here to view the region in the $x-y$ plane, which comprises the domain for the integral, having $y$ playing the role of the independent variable and having $x$ as a function of $y$.
The region in the $x-y$ plane is bounded from $x=y^2-y$ to $x=y$ as $y$ goes from $0$ to $2$.
Therefore, the volume integral is
$$V=\int_0^2\int_{y^2-y}^y(3x^2+y^2)dx\,dy$$
Can you finish?


TITLE: Cumulative distribution function on uniform distribution
QUESTION [1 upvotes]: For $l>0$, $X$ is a random variable according to the uniform distribution on the interval $[0, l].$
At this time, the probability density function $f(x)$ of $X$ is
$f_X(x)=  \cases{0 & if $x<0$\\
        \frac{1}{l} & if $ 0 \le x \le l$\\
      0 & if $x > l$}
$
The length of the shorter interval obtained by dividing the following  interval $[0,l]$ into 2 at random  is expressed by a random variable $Y$.
The length of the longer interval obtained by randomly dividing  interval $[0,l]$ into two parts is expressed by a random variable $Z$.
Find the distribution function $F_y $ of Y. Find the distribution function
$ F_Z$ of Z.
$f_Y(x)=  \cases{0 & if $x<0$\\
        \frac{1}{l/2} & if $ 0 \le x <\frac{l}{2}$\\
      0 & if $x \le l$}
$
$f_Z(x)=  \cases{0 & if $x<0$\\
\frac{1}{l/2} & if $ 0 \le x <1/2$\\
        \frac{4}{(2-l)l} & if $ l/2 \le x <1$\\
      0 & if $x \le l$}
$
I'm not sure, am I correct? And after this, do I just need to integrate it to find $F_Y$?

REPLY [1 votes]: You are not quite correct, but not far away.
Apart from your typographical confusion between $1$ and $l$, you have some inequalities the wrong way round and some major issues in the density function for $Z$: it cannot be less than $l/2$ since it is the longer of the two pieces, and in fact it has a uniform distribution between $l/2$ and $l$. I think you should have something like
$f_Y(x)=  \cases{0 & if $x<0$\\
        \frac{1}{l/2} & if $ 0 \le x \le \frac{l}{2}$\\
      0 & if $x \gt l/2$}
$
$f_Z(x)=  \cases{0 & if $x<0$\\
0 & if $ 0 \le x \le l/2$\\
        \frac{1}{l/2} & if $ l/2 \lt x \le l$\\
      0 & if $x \gt l$}
$
Then, as you suggest, you should integrate to get the cumulative distribution functions


TITLE: If $f_n{\rightarrow} f $ uniformly and $f$ is bounded then prove that $f_n^{2}{\rightarrow}f^{2} $ uniformly
QUESTION [0 upvotes]: If $f_n{\rightarrow} f $ uniformly and $f$ is bounded, how can we prove that $f_n^{2}{\rightarrow}f^{2} $ uniformly? I am revising for my analysis final and got stuck on this problem, any help is really appreciated.

REPLY [3 votes]: You can use the fact that $f$ is bounded and $f_n\to f$ uniformly to show that there is a uniform bound $M>0$ such that $|f_n(x)|<M$ and $|f(x)|<M$ for all $n$ and all $x$.
You can use the identity $a^2-b^2 = (a+b)(a-b)$ to bound the difference of squares in terms of $M$ and the difference of the original functions.

REPLY [0 votes]: Hint: $f_n^2-f^2=(f_n-f)(f_n+f)$.


TITLE: Standard Interpolation between Bochner spaces
QUESTION [1 upvotes]: I've read the following in a few papers:
Given: Let $\Omega \subset R^d$ be a Lipschitz domain. A sequence $f_n$ converges strongly to $f$ in $L^2(0,T;L^2(\Omega))$ and
weakly in $L^2(0,T;H^2(\Omega))$.
Then by a standard interpolation argument it follows that
$f_n$ converges strongly to $f$ in $L^2(0,T;H^1(\Omega))$.
Now my question is what is meant by standard interpolation argument?
Thanks for your answers!

REPLY [0 votes]: You may want to have a look for a book on interpolation theory, there is, e.g., a great introduction by Tartar.
There are, typically, two interpolation results:
$$\|u\|_{L^p} \le \|u\|_{L^r}^\mu \, \|u\|_{L^s}^{1-\mu},$$
(Lebesgue scale), if $1/p = \mu/r + (1-\mu)/s$ and
$$\|u\|_{H^1} \le C \, \|u\|_{L^2}^{1/2} \, \|u\|_{H^2}^{1/2}$$ (Sobolev scale).
Your proof requires the second one (use boundedness in $H^2$).


text:
\begin{document}
\date{}
\maketitle

\begin{abstract}
In this paper we discuss a conjecture on intermediate subfactors
which is a generalization of Wall's conjecture from the theory of
finite groups.  We explore special cases of this conjecture and
present supporting evidence.  In particular we prove special cases
of this conjecture related to some finite dimensional Kac Algebras
of Izumi-Kosaki type which include relative version of Wall's
conjecture for solvable groups.

\end{abstract}

\newpage

\section{Introduction}

Let $M$ be a factor represented on a Hilbert space and $N$ a
subfactor of $M$ which is irreducible, i.e.,$N'\cap M= \C$. Let $K$
be an intermediate von Neumann subalgebra for the inclusion
$N\subset M.$ Note that $K'\cap K\subset N'\cap M = \C,$ $K$ is
automatically a factor. Hence the set of all intermediate subfactors
for $N\subset M$ forms a lattice under two natural operations
$\wedge$ and $\vee$ defined by:
\[
K_1\wedge K_2= K_1\cap K_2, K_1\vee K_2= (K_1\cup K_2)''.
\]
The commutant map $K\rightarrow K'$ maps an intermediate subfactor
$N\subset K\subset M$ to $M'\subset K'\subset N'.$ This map
exchanges the two natural operations defined above.

Let $M\subset M_1$ be the Jones basic construction of $N\subset M.$
Then  $M\subset M_1$ is canonically isomorphic to $M'\subset N'$,
and the lattice of intermediate subfactors for $N\subset M$  is
related to the lattice of intermediate subfactors for $M\subset M_1$
by the commutant map defined as above.

 Let $G_1$ be a group and $G_2$ be a subgroup of $G_1$. An
interval sublattice $[G_1/G_2]$ is the lattice formed by all
intermediate subgroups $K, G_2\subseteq K\subseteq G_1.$

By cross product construction and Galois correspondence,  every
interval sublattice of finite groups can be realized as intermediate
subfactor lattice of finite index. Hence the study of intermediate
subfactor lattice of finite index is a natural generalization of the
study of interval sublattice of finite groups. The study of
intermediate subfactors has been very active in recent years(cf.
\cite{BJ},\cite{GJ}, \cite{JXu},\cite{Jh2}, \cite{ILP},
\cite{Longo4}, \cite{Wat} and \cite{ SW} for only a partial list).
By a result of S. Popa (cf. \cite{Popa}), if a subfactor $N\subset
M$ is irreducible and has finite index, then the set of intermediate
subfactors between $N$ and  $M$ is finite. This result was also
independently proved by Y. Watatani (cf. \cite{Wat}). In \cite{Wat},
Y. Watatani investigated the question of which finite lattices can
be realized as intermediate subfactor lattices. Related questions
were further studied by P. Grossman and V. F. R. Jones in \cite{GJ}
under certain conditions. As emphasized in \cite{GJ}, even for a
lattice consisting of six elements with shape a hexagon, it
is not clear if it can be realized as intermediate subfactor lattice
with finite index. This question has been solved recently by M.
Aschbacher in \cite{Asch} among other things. In \cite{Asch}, M.
Aschbacher  constructed a finite group $G_1$ with a subgroup $G_2$ such
that the interval sublattice $[G_1/G_2]$ is a hexagon. The lattices
that appear in \cite{GJ,Wat,Asch} can all be realized as interval
sublattice of finite groups. There are a number of old problems
about interval sublattice of finite groups. It is therefore a
natural programme to investigate if these old problems have any
generalizations to subfactor setting. The hope is that maybe
subfactor theory can provide new perspective on these old
problems.\par In \cite{Xl} we consider the problem whether the very
simple lattice $M_n$ consisting of a largest, a smallest and $n$
pairwise incomparable elements can be realized as subfactor lattice.
We showed in \cite{Xl} all $M_{2n}$ are realized as the lattice of
intermediate subfactors of a pair of hyperfinite type $III_1$
factors with finite depth. Since it is conjectured that infinitely
many $M_{2n}$ can not be realized as interval sublattices of finite
groups (cf. \cite{Balu} and \cite{Palfy}), our result shows that if
one is looking for obstructions for realizing finite lattice as
lattice of intermediate subfactors with finite index, then the
obstruction is very different from what one may find in finite group
theory.\par

In 1961 G. E. Wall conjectured that the number of maximal subgroups
of a finite group $G$ is less than $|G|$, the order of $G$ (cf.
\cite{wall61}). In the same paper he proved his conjecture when $G$
is solvable. See \cite{lie} for more recent result on Wall's
conjecture.

Wall's conjecture can be naturally generalized to a conjecture about
maximal elements in the lattice of intermediate subfactors. What we
mean by maximal elements are those subfactors $K\neq M,N $ with the
property that if $K_1$ is an intermediate subfactor and $K\subset
K_1,$ then $K_1=M$ or $K.$ Minimal elements are defined similarly
where $N$ is not considered as an minimal element. When $M$ is the
cross product of $N$ by a finite group $G$, the maximal elements
correspond to maximal subgroups of $G,$ and the order of $G$ is the
dimension of second higher relative commutant. Hence a natural
generalization of Wall's conjecture as proposed in \cite{X1} is the
following:
\begin{conjecture}\label{wall}
Let $N\subset M$ be an irreducible subfactor with finite index. Then
the number of maximal  intermediate subfactors is less than
dimension of $N'\cap M_1$ (the dimension of second higher relative
commutant of $N\subset M$).
\end{conjecture}
We note that since maximal intermediate subfactors in $N\subset M$
correspond to minimal intermediate subfactors in $M\subset M_1,$ and
the dimension of second higher relative commutant remains the same,
the conjecture  is equivalent to a similar conjecture as above with
maximal replaced by minimal.\par In \cite{X1}, Conjecture \ref{wall}
is verified for subfactors coming from certain conformal field
theories. These are subfactors not related to groups in general. In
this paper we consider those subfactors which are more closely
related to groups and more generally Hopf algebras.\par

If we take $N$ and $M$ to be cross products of a factor $P$ by $H$
and $G$ with $H$ a subgroup of $G$, then  the minimal version of
conjecture \ref{wall} in this case states that the number of minimal
subgroups of $G$ which strictly contain $H$ is less than the number
of double cosets of $H$ in $G$. This follows from simple counting
argument. The nontrivial case is the maximal version of the above
conjecture. In this case it gives a generalization of Wall's
conjecture which we call relative version of Wall's conjecture. The
relative version of Wall's conjecture states that the number of
maximal subgroups of $G$ strictly containing a subgroup $H$ is less
than the number of double cosets of $H$ in $G.$  As a simple example
when this can be proved, consider $G=H\times H,$ and $D \cong H$ is a
diagonal subgroup of $G.$ Then the set of maximal subgroups of $G$
containing $D$ are in one to one correspondence with the set of
maximal normal subgroups of $H,$ and it is easy to check that the set
of maximal normal subgroups has cardinality less than the number of
irreducible representations of $H.$ On the other hand  the number of
double cosets of $H$ in $G$ is the same as the number of conjugacy
classes of $H,$ and this is the same as the number of irreducible
representations of $H.$ So we have proved the relative version of
Wall's conjecture in this case.

In \S\ref{relativesection} we will prove this relative version of
Wall's conjecture for  $G$ solvable.   We
will present two proofs. The first  proof is motivated by an idea of
V. F. R. Jones which is to seek linear independent vectors
associated with minimal subfactors in the space of second higher
relative commutant. This proof is indirect but we hope that the idea
will prove to be useful for more general case. We formulate a
conjecture for general subfactors (cf. Conjecture \ref{mod}) which
is stronger than Conjecture \ref{wall}, and for solvable groups this
conjecture is proved in \cite{X1}. Here we modify the proof in
\cite{X1} to prove a linear independence result (cf. Th. \ref{sol}),
and this result implies the relative Wall conjecture for solvable groups.
 The second proof is a
more direct proof using properties of maximal subgroups of solvable groups.
\par The cross
product by finite group subfactor  is a special case of depth $2$
subfactor. If we take $N\subset M$ to be depth 2, by
\cite{David},\cite{NV} such a subfactor comes from cross product by
a finite dimensional *-Hopf algebra or Kac algebra $\A$. By
\cite{ILP} or \cite{NV} the intermediate subfactors are in one to
one correspondence to the set of left (or right) coideals of $\A$.
Then Conjecture \ref{wall} states that the number of maximal (resp.
minimal) right coideals of $\A$ is less than the dimension of $\A$.
In \S\ref{iksection} we will prove this for the case of Kac algebras
$\A$ of Izumi-Kosaki type with solvable groups as considered in
\cite{IK}. We also prove Conjecture \ref{wall} for the intermediate
subfactors of Izumi-Kosaki type with solvable groups as considered
in \cite{IK} which are not necessarily of depth 2 (cf. Th.
\ref{relative2}). Th. \ref{relative2} generalizes Th.
\ref{relativeth}. It is interesting to note that the same type of
first cohomology problem encountered in Remark \ref{coh} also
appears here but in a different way and solvability is once again
used to ensure that the first cohomology group is trivial (cf. Lemma
\ref{count1} and Lemma \ref{count2}).\par We note that recently
lattices of intermediate for other types of Kac algebras have been
obtained in \cite{DT}. Our conjecture can be verified in the
examples of \cite{DT} where complete lattice of intermediate
subfactors are determined. The maximal (or minimal) coideals are
very few compared with the dimension of the Kac algebra in these
examples of \cite{DT}.
\par

In \S\ref{tensorsection} we first present a  lemma which bounds the
number of maximal subgroups of a group $X\times Y$ which does not
contain either $X$ nor $Y.$ This lemma gives a proof of Wall's
conjecture for $X\times Y$ assuming that Wall's conjecture is true
for $X$ and $Y.$ We then propose a natural conjecture about tensor
products of subfactors.

At the end of this introduction let us consider a fusion algebra
version of Conjecture \ref{wall}. Let $\rho_i\in \End(M), i=1,...,n$
be a finite system of irreducible sectors of a properly infinite
factor $M$ which is closed under fusion. Consider the Longo-Rehren
subfactor associated with such a system (cf. \cite{LR}). By
\cite{I}, the intermediate subfactors are in one to one
correspondence with the fusion subalgebras which are generated by a
subset of simple objects $\rho_i$ , and Conjecture \ref{wall} states
that the number of such maximal fusion subalgebras is bounded by $n$
which is the number of simple objects. This motivates us to make the
following conjecture:
\begin{conjecture}\label{fusioncase}
Let $\F$ be a finite dimensional semisimple fusion algebra with $n$
simple objects. Then  the number of maximal fusion subalgebras which
are generated by a subset of the simple objects of $\F$ is less than
$n$.
\end{conjecture}
If we take $\F$ to be the group algebra of $G,$ then Conjecture
\ref{fusioncase} is equivalent to Wall's conjecture.\par

If we take $\F$ to be the fusion algebra of representations of a
finite group $G$, then the maximal fusion subalgebras are in one to
one correspondence to minimal normal subgroups of $G,$ and the
number of such subgroups are less than the number of conjugacy
classes of $G$, which is the same as the number of simple objects of
$\F$. This is a special case of a more general result of   D.
Nikshych and V. Ostrik, who prove that Conjecture \ref{fusioncase}
is true for commutative $\F$ \cite{NO}.

The second named author (F.X.) would like to thank  Prof. V. F. R.
Jones for his encouragement and useful comments on conjecture
\ref{wall} which inspired our first proof presented in
\S\ref{relativesection}, and for communications on numerical
evidence supporting the relative version of Wall's conjecture. F.X.
would also like to thank Prof. D. Bisch for invitation to a
conference on subfactors and fusion categories in Nashville where
some of the results of this paper were discussed, and Professors
Marie-Claude David,  D. Nikshych and V. Ostrik for useful
communications.
\section{Relative version of Wall's
conjecture for solvable groups}\label{relativesection}

In this section we will prove Theorem \ref{relativeth}, which
confirms the relative version of Wall's conjecture for solvable
groups. We will give two proofs of this result. The first proof is
motivated by the following  conjecture, formulated as Conjecture A.1
in \cite{X1}, which can be stated for general subfactors:

\begin{conjecture}\label{mod}
Let $N\subset M$ be an irreducible subfactor with finite Jones
index, and let $P_i, 1\leq i\leq n$ be the set of minimal
intermediate subfactors. Denote by $e_i\in N'\cap M_1, 1\leq i\leq
n$ the Jones projections $e_i$ from $M$ onto $P_i$ and $e_N$ the
Jones projections $e_N$ from $M$ onto $N.$ Then there are vectors
$\xi_i, \xi\in N'\cap M_1$ such that $e_i \xi_i=\xi_i,1\leq i\leq n,
e_N\xi=\xi,$ and $\xi_i, 1\leq i\leq n,\xi$ are linearly
independent.
\end{conjecture}

\begin{remark}
We note that unlike conjecture \ref{wall}, the conjecture above
makes use of the algebra structure of $N'\cap M_1$ and therefore
does not immediately imply the dual version or if one replaces
minimal by maximal.
\end{remark}

By definition conjecture \ref{mod} implies conjecture \ref{wall}. In
the case of subfactors from groups,  it is easy to check that
conjecture \ref{mod} is equivalent to:

\begin{conjecture}\label{mod2'}
Let $K_i, 1\leq i\leq n$ be a set of maximal subgroups of $G.$ Then
there are vectors $\xi_i\in l(G), 1\leq i\leq n$ such that $
e_G\xi_i=0,$ $\xi_i$ are $K_i$ invariant and linearly independent.
\end{conjecture}

This conjecture is proved in \cite{X1}  when $G$ is solvable. It
turns out a modification of the proof presented in \cite{X1} gives a
proof of a stronger statement. Let us make the following stronger
conjecture.   First we need to introduce some notation.
If $H$ is a subgroup of $G$, let $\ell(H)=\ell(G,H)$ be the permutation
module $\bbc_H^G$.  Let $\ell_0(H)$ denote the hyperplane of
weight zero vectors in $\ell(H)$ (i.e. the complement to the $1$-dimensional
$G$-fixed space on $\ell(H)$).

\begin{conjecture}\label{mod2}
Let $K_i, 1\leq i\leq n$ be a set of maximal subgroups of $G.$
Set $H = \cap K_i$.  Then
there are vectors $\xi_i\in \ell_0(H), 1\leq i\leq n$  that
 are $K_i$-invariant and linearly independent.   In particular,
 this implies that $n \le \dim \ell_0(H)^H < |H / G \backslash H|$.
\end{conjecture}

We will prove Conjecture \ref{mod2} for solvable groups by modifying
the arguments of \cite{X1}.       We begin with some preparations
that hold for all finite groups.

\begin{lemma}\label{prim}
Suppose that $K_1, \ldots, K_n$ are conjugate maximal subgroups
of the finite group $G$.   Then Conjecture \ref{mod2} holds for
$\{K_1, \ldots, K_n\}$.
 \end{lemma}

\prf  Set $K=K_1$.  If $n=1$, the result is obvious (in particular, if $K$ is normal in $G$).  So assume that $n > 1$.  Let $H = \cap K_i$.
Of course, $\ell_0(K)$  is a submodule of $\ell_0(H)$.    Let $K_1, \ldots, K_m, m \ge n$ be the set of
all conjugates of $K$.  Since $K$ is not normal in $G$,  $K$ is self normalizing whence
if we choose a permutation basis $\{ v_i   | 1 \le i \le m\}$ for $\ell(K)$, then the stabilizers of the $v_i$ are precisely the $K_i$.
If $m > n$, then the vectors $v_i - v_0 \in \ell_0(K), 1 \le i \le n$ are clearly linearly independent (here $v_0 = \sum v_i$ is fixed
by $G$).   So it suffices to assume that $m=n$ and so in particular, $H$ is normal in $G$.  So we may assume
that $H=1$.   Let $V$ be a nontrivial irreducible submodule of $\ell_0(K)$.   Then $K$ does not act trivially on $V$.
Note that, by Frobenius reciprocity,  the multiplicity of $V$ in $\ell_0(K)$ is precisely $\dim V^K  < \dim V$.  Of course
$\dim V$ is the multiplicity of $V$ in $\ell_0(H)$.
  Thus,  $\ell_0(K) \oplus V$ is a submodule of $\ell_0(H)$.
  Now choose vectors $v_i - v_0, 1 \le i < m$  as above and $w_m$ any fixed vector of $K_m$ in $V$.  These are
obviously linearly independent.
 \qed

 \par
 Next we prove a reduction theorem for Conjecture \ref{mod2}.   Note that the reduction depends on
 the existence of the vectors and not just on cardinality.

\begin{lemma}\label{induction}   Let $\mathcal{S}$ be a family of finite simple groups.
Let $\mathcal{F(S)}$ denote the family of all finite groups with all composition factors
in $\mathcal{S}$.   Let $K_1, \ldots, K_n$ be maximal subgroups of the finite group
$G$ in $\mathcal{F(S)}$ and assume that Conjecture \ref{mod2} fails with $n|G|$ minimal.
Then each $K_i$ has trivial core in $G$.  In particular, $G$ is a primitive permutation group.
\end{lemma}

\prf   Suppose that $N$ is a nontrivial normal subgroup of $G$ contained in $K_1$.
Set $H = \cap K_i$.
If each $K_i$ contains $N$,  then $\ell_0(H)$ is a $G/N$-module and so
$G/N, K_1/N, \ldots, K_n/N$ would give a counterexample to the conjecture.

Reorder the $K_i$ so that $N \le K_i$  if and only if $i \le s < n$.   Note that $NK_j=G$ for
$j > s$.   By the minimality of $|G|n$,  we can choose  $v_1, \ldots, v_n  \in \ell_0(H)$
with $K_jv_j=v_j$ for  all $j$ such that $\{v_1, \ldots, v_s\}$ and $\{v_{s+1}, \ldots, v_n\}$
are linearly independent.    It thus suffices to show that spans of $v_1, \ldots, v_s$ and
$v_{s+1}, \ldots, v_n$ have trivial intersection.  Suppose that $u$ is in this intersection.

Since $e_Ne_{K_j}=e_G$ for $j > s$ (since $G=NK_j$), it follows that $0 = e_Gv_j = e_Ne_{K_j}v_j=e_Nv_j$
for $j > s$.  Thus, $e_Nu=0$.  Since $N$ fixes $v_i, i \le s$, it follows that $e_Nu=u$.  Thus, $u=0$
and the result follows.
\qed

\begin{theorem}\label{sol}
Conjecture \ref{mod2} is true for $G$ solvable.
\end{theorem}

\prf   Consider a counterexample with $|G| n$ minimal.  By Lemma
\ref{induction}, none of the $K_i$ contain a normal subgroup.  It
follows that $G$ is a solvable primitive permutation group, whence
$G = AK$ where $A$ is elementary abelian and $K$ acts irreducibly on
$A$. In particular, any maximal subgroup of $G$ either contains $A$
or is a complement to $A$. Since the core of each $K_i$ is trivial,
 $G=AK_i$ for each $i$.
Since $G$ is solvable, $H^1(K,A)=0$, whence all of the $K_i$ are conjugate.
Now apply Lemma \ref{prim} to complete the proof.
\qed

\begin{remark}\label{coh}  As we have seen, a minimal counterexample
to Conjecture 2.4 would be a
primitive permutation groups, and the set of maximal subgroups must all have
trivial core. Such groups are classified by Aschbacher-O'Nan-Scott
theorem (cf. \S 4 of \cite{NS}). The first case is when  $G$ is the
semidirect product of an elementary abelian group $V$ by $K_1,$ and
the action of $K_1$ on $V$ is irreducible. When $G$ is not solvable,
maximal subgroups $K$ of $G$ with trivial core are not conjugates of
$K_1$, and our proof as above does not work. Such maximal subgroups
are related to the first cohomology of $K_1$ with coefficients in
$V,$ and conjecture \ref{mod2} implies that the order of this
cohomology is less than $|K_1|$ (cf. Question 12.2 of \cite{lub}).
Unfortunately even though it is believed that the order of this
cohomology is small (cf. \cite{Gu}), the bound $|K_1|$ has not been
achieved yet.
\end{remark}

We give a second proof of Conjecture \ref{mod2} for solvable groups which
is not inductive.

Let $G$ be a solvable group.  Let  $H \le G$ and let   $K_1, \ldots, K_r$
denote a maximal collection of maximal subgroups of $G$ containing $H$ which are not conjugate.
Let $K_{ij}, 1 \le i \le r, 1 \le j \le n_i$ denote
the set of all maximal subgroups of $G$ containing $H$ where $K_{ij}$ is conjugate to $K_i$.

It is easy to see that $G=K_iK_j$ for $i \ne j$ (cf \cite{AsbG}).    Thus, $\Hom(\ell_0(K_i), \ell_0(K_j)=0$
if $i \ne j$.   If $K_i$ is normal in $G$, set $V_i=0$.  If not, let $V_i$ be a nontrivial irreducible submodule of $\ell_0(K_i)$
such that $\ell_0(K_i) \oplus V_i$ embeds in $\ell_0(H)$ (as in the proof of Lemma \ref{prim}).
Thus $X:= \oplus_i (W_i \oplus V_i)$ embeds
in $\ell_0(H)$ and as above, we can choose $v_{ij}$ in $W_i \oplus V_i$ linearly
independent with $K_{ij}$ the stabilizer of $v_{ij}$.
\qed

Of course, this gives:

\begin{theorem} \label{relativeth} Let $G$ be a finite solvable group.  Let $H$ be a subgroup of $G$.
Then the number of maximal subgroups of $G$ which contain $H$ is less than
$|H/G \backslash H|$.
\end{theorem}

\section{Kac algebras of Izumi-Kosaki type for solvable
groups}\label{iksection} In this section we will prove Conjecture
\ref{wall} for Kac algebras of  Izumi-Kosaki type for solvable
groups. These Kac algebras are introduced in \cite{IK} and in more
details in \cite{IK2} by considering compositions of group type
subfactors. Let us first recall some definitions from \cite{IK} to
set up our notations. The reader is refereed to \cite{IK2} for more
details. Let $G=N\rtimes H$ be semidirect product of two finite
groups $N,H.$ For $n\in N, h\in H,$ we define $n^h:=h^{-1}nh.$
Denote by $L(N)$ the set of complex valued functions on $N.$ For
$f\in L(N), f^h(n):=f(h^{-1}nh), h\in H.$
\begin{definition}\label{cocycle1}
Denote by $\eta_h(n_1,n_2), \xi_n(h_1,h_2)$ $U(1)$ valued cocycles
as defined in \S2 of \cite{IK} which verify the following cocycle
conditions:
$$
\eta_h(n_1,n_2)\eta_h(n_1n_2,n_3)=\eta_h(n_1,n_2n_3)\eta_h(n_2,n_3),
\xi_n(h_1h_2,h_3)\xi_n(h_1,h_2)=\xi_n(h_1,h_2h_3)\xi_n(h_2,h_3);
$$
Moreover, these cocycles verify the following Pentagon equation:
$$
\frac{\eta_{h_1}(n_1,n_2)\eta_{h_2}(n_1^{h_1},n_2^{h_2})}{\eta_{h_1}(n_1,n_2)}
=\frac{\xi_{n_1n_2}(h_1,h_2)}{\xi_{n_1}(h_1,h_2)\xi_{n_2}(h_1,h_2)}
$$
and normalizations:
$$
\eta_h(e,n_2)=\eta_h(n_1,e)=\xi_n(e,h_2)=\xi_n(h_1,e)=\eta_e(n_1,n_2)=1.
$$

\end{definition}
For subfactor motivations for introducing these cocycles, we refer
the reader to \S2 of \cite{IK}.
\begin{definition}\label{ikalg}
Kac algebras of Izumi-Kosaki  type are defined as Hopf algebras
$\A=L(N)\rtimes_\xi H$ whose Hopf algebra structures are given in
\cite{IK} as follows:\par (1) Algebra products:
$$
(f_1(n),h_1)(f_2(n),h_2)=(f_1(n)f_2^{h_1}(n)\xi_n(h_1,h_2),h_1h_2)
$$
where $f_2^{h_1}(n):=f_2(h_1^{-1}nh_1);$
\par
(2) Coproducts:
$$ \Delta
(n,h)=\sum_{n_2}\eta_h(nn_2^{-1},n_2)(nn_2^{-1},h)\otimes(n_2,h)
$$
\par
(3) $*$ structure:
$$
(f,h)^*= (\overline{f\xi(h,h^{-1})}^{h^{-1}}, h^{-1})
$$
\end{definition}
The following two operators on $L(N)$ will play an important role:
\begin{definition}\label{leftright}
$$
(L_{n,\eta_h}f)(m):=f(nm)\eta_h(n,m),
(R_{n,\eta_h}f)(m):=f(mn)\eta_h(n,m), \forall n,m\in N,h\in H.
$$
\end{definition}
The following lemma summarize the properties of these operators
which follow from definitions:
\begin{lemma}\label{lrprop}
$$
L_{n_1,\eta_h}L_{n_2,\eta_h}=L_{n_2n_1,\eta_h}\eta_h(n_2,n_1),
R_{n_1,\eta_h}R_{n_2,\eta_h}=R_{n_1n_2,\eta_h}\eta_h(n_1,n_2),
L_{n_1,\eta_h}R_{n_2,\eta_h}=R_{n_2,\eta_h}L_{n_1,\eta_h}.
$$
\end{lemma}
The subfactor associated with $\A$ is of the form $\L^\A\subset \L$
where $\L^\A$ is the fixed point subfactor of a factor $\L$ under
the action of $\A$ as defined in \S4 of \cite{ILP}. By \cite{ILP},
any intermediate subfactor of $\L^\A\subset \L$ is of the form
$\L^\B\subset L,$ where $\B$ is a right coideal of $\A,$ i.e., an
$*$ subalgebra of $\A$ which is verifies that $\Delta (\B)\subset
\B\otimes \A.$

The following theorem gives a characterization of  coideals of $\A$:
\begin{theorem}\label{coideal}
Let $\B$ be a right (resp. left) coideal of $\A$. Then there are
subgroups $H_1\leq H, N_1\leq N$ and $U(1)$ valued function
$\lambda:N_1\times H_1\rightarrow U(1)$ such that:\par (1) $\forall
h\in H_1, hN_1h^{-1}=N_1;$\par
\begin{equation}\label{lambdaeq}
\lambda(n_1,h)\lambda(n_2,h)=\lambda(n_1n_2,h)\eta_h(n_1,n_2),
\lambda(n,h_1)\lambda(h_1^{-1}nh_1,h)\xi_n(h_1,h_2)=\lambda(n,h_1h_2)
\end{equation}
and $\B=\oplus_{h\in H_1}(C(h),h)$ where each $C(h)\subset L(N)$
consists of functions $f\in L(N)$ such that
$L_{n,\eta_h}f=\lambda(n,h)f$ (resp. $R_{n,\eta_h}f=\lambda(n,h)f$)
$\forall n\in N_1, h\in H_1$.\par Conversely, any triple
$(N_1,H_1,\lambda)$ which verify the above conditions uniquely
determine a coideal of $\A.$
\end{theorem}
\prf We will prove the theorem for the case when $\B$ is a right
coideal of $\A$. The remaining case is similar. We write elements of
$\B$ as $\sum_h (f_h,h)$ where $f_h\in L(N).$ We have
$$
\Delta \sum_h(f_h,h)= \sum_{n_2,h}
(R_{n_2,\eta_h}f_h,h)\otimes(n_2,h)
$$
Since $\B$ is a right coideal, it follows that for each fixed
$(n_2,h)$, $(R_{n_2,\eta_h}f_h,h)\in \B.$ So we have
$\B=\oplus_h(C(h),h)$ with $C(h)$ a subspace of $L(N)$ which is
mapped by $R_{n,h}$ to itself. Since $\B$ is also an algebra, we
have
\begin{equation}\label{cs}
C(h_1)C(h_2)^{h_1}\xi(h_1,h_2)\subset C(h_1h_2)\\
\end{equation}
In particular $C(e)$ is a subalgebra of $L(N)$ which affords a right
representation of $N.$ It follows that there is a subgroup $N_1\leq
N$ such that $C(e)$ is the space of $N_1$-left invariant functions
on $N.$ Let $N=\bigcup_i N_1b_i, 1\leq i\leq k$ with $k=|N|/|N_1|$
be the left coset decompositions of $N.$ Then $\delta_{Nb_i}$ is a
basis of $C(e).$

\par
 Since $\B$ is an $*$ algebra, it follows that if
$(f_h,h)\in \B,$ then
$$(f_h,h)^*\in
(C(h^{-1}),h^{-1}),(f_h,h)(g_h,h)^*= (f_h\bar{g_h},e)\in (C(e),e).$$
Let $H_1:=\{h\in H| C(h)\neq 0\}.$ It follows easily from above that
$H_1$ is a subgroup of $H.$ By equation (\ref{cs}) $C(e)C(h)\subset
C(h).$ so it follows that $C(h)=\oplus_i C(h)\delta_{N_1b_i}.$
Assume that $h\in H_1$ so $C(h)\neq 0.$ Since $R_{b_i^{1},\eta_h}$
maps $C(h)$ to itself, we can assume that $C(h)\delta_{N_1}\neq 0.$
Let $f_i\neq 0\in C(h)\delta_{N},i=1,2$ then $f_1\bar f_2\in
C(e)\delta_{N_1}=\C\delta_{N_1}.$ We conclude that
$C(h)\delta_{N_1}$ is one dimensional, and by using operator
$R_{b_i^{-1},\eta_h},$ we conclude that $C(h)\delta_{N_1b_i}$ is one
dimensional. Let $f_h\neq 0\in C(h)\delta_{N_1},$ then since
$R_{n_1,\eta_h}$ maps $C(h)\delta_{N_1}$ to itself, there is a
function $\lambda:N_1\times H_1\rightarrow U(1)$ such that
$$
R_{n_1,\eta_h}f_h=\lambda(n_1,h)f_h.
$$
We can assume that $f_h(e)=1.$ Then we have
$$
(R_{n_1,\eta_h}f_h)(e)=\lambda(n_1,h)=f_h(n_1).
$$
Equation (\ref{lambdaeq}) follows from Lemma \ref{lrprop} and
equation (\ref{cs}). Let us show that $C(h),h\in H_1$ is the
subspace of $L(N)$ which verifies
$$
L_{n_1,\eta_h}f=\lambda(n_1,h)f.
$$
We note that by definition \ref{leftright} $f_h(n_1)=\lambda(n_1,h)$
verifies
$$
(L_{n_1,\eta_h}f_h)(m)=\lambda(n_1m,h)\eta_h(n_1,m)=\lambda(n,h)\lambda(m,h)
=\lambda(n,h)f_h(m)
$$
where in the last equation we have used equation (\ref{lambdaeq}).
Since $C(h)$ is the linear span of $R_{n,\eta_h}f_h$, by lemma
\ref{lrprop} we have proved that for any $f\in C(h),
L_{n_1,\eta_h}f=\lambda(n_1,h)f.$ On the other hand by counting
dimensions we conclude that $C(h),h\in H_1$ is the subspace of
$L(N)$ which verifies
$$
L_{n_1,\eta_h}f=\lambda(n_1,h)f.
$$
Let us show that $\forall h\in H_1, hN_1h^{-1}=N_1.$ By equation
(\ref{cs}) we have $f_h(n_1)\delta_{N_1}^h\subset C(h)\delta_{N_1}$
which is one dimensional, and it follows that
$\delta_{N_1}^h=\delta_{N_1}$ for all $h\in H_1,$ i.e.,
$hN_1h^{-1}=N_1.$\par

Conversely for  any triple $(N_1,H_1,\lambda)$ which verify the
conditions in theorem \ref{coideal}, we can simply define
$\B:=\oplus_{h_1\in H_1}(C(h),h)$ where $C(h)\subset L(N)$ consists
of functions $f\in L(N)$ such that $L_{n,\eta_h}f=\lambda(n,h)f.$ We
need to check that $\B$ is a right coideal. By inspection it is
enough to check equation (\ref{cs}). By definition we need to check
that if $f_{h_i}\in C(h_i), i=1,2,$ then
$g(n):=f_{h_1}(n)f_{h_2}^{h_1}(n)\xi_n(h_1,h_2)\in C(h_1h_2).$ So we
need to show that $(L_{n,\eta_{h_1h_2}}g)(m)=\lambda(n,h_1h_2)g(m),
\forall n\in N_1, m\in N.$ Since $f_{h_i}\in C(h_i), i=1,2,$ we have
$$
f_{h_i}(nm)\eta_{h_i}(n,m)=\lambda(n,h_1)f_{h_i}(m),i=1,2.
$$
By using above equation and equation (\ref{lambdaeq}) it follows
that $(L_{n,\eta_{h_1h_2}}g)(m)=\lambda(n,h_1h_2)g(m), \forall n\in
N_1, m\in N$ iff the following holds:
$$
\frac{\eta_{h_1}(n,m)\eta_{h_2}(n^{h_1},m^{h_2})}{\eta_{h_1h_2}(n,m)}
=\frac{\xi_{nm}(h_1,h_2)}{\xi_{n}(h_1,h_2)\xi_{m}(h_1,h_2)}
$$

which is the pentagon equation in definition \ref{cocycle1}.

\qed

For a coideal $\B$ with $(N_1,H_1,\lambda)$ as in Th. \ref{coideal},
we shall refer to $(N_1,H_1,\lambda)$ as {\it the triple} associated
with $\B.$ We note that by Th. \ref{coideal}, such triple uniquely
determine $\B.$ Moreover, suppose that the triples associated with
$\B_i$ are given by $(N_i,H_i,\lambda_i), i=1,2.$ Then $\B_1\subset
\B_2$ iff $N_1\supset N_2, H_1\subset H_2$ and $\lambda_1,\lambda_2$
agree on $N_2\times H_1.$
\begin{lemma}\label{count1}
Let $\B$ be a right coideal of $\A$ as in Theorem \ref{coideal} with
triple $(N_1,H_1, \lambda).$ The the number of right coideals of
$\A$ with the same $(N_1,H_1)$ are given as follows: Let $\hat{N_1}$
be the set of homomorphisms from $N_1$ to $U(1)$ and form a group
$\hat{N_1}\rtimes H_1$. Then the  right coideals of $\A$ with the
same $(N_1,H_1)$ are in one to one correspondence with the set of
cocycles from $H_1$ to $\hat{N_1},$ i.e., maps $\mu: H_1\rightarrow
\hat{N_1}$ such that
$$
\mu(h_1)\mu(h_2)^{h_1}=\mu(h_1h_2).
$$
\end{lemma}
\prf Let $\B_1$ be a right coideal of $\A$ with triple
$(N_1,H_1,\lambda_1)$ as in Theorem \ref{coideal}. Let
$\mu:=\lambda_1/\lambda.$ By equation (\ref{lambdaeq}) we conclude
that $\mu$ is a cocycle from $H_1$ to $\hat{N_1}.$ Conversely, if
$\mu$ is a cocycle from $H_1$ to $\hat{N_1},$ then $\B_1$ associated
with the triple $(N_1,H_1,\lambda \mu)$ is a  right coideal of $\A$
by Theorem \ref{coideal}. \qed
\begin{lemma}\label{count2}
Let $H_1\subset H, N_1\neq \{e\}\subset N$ such that
$hN_1h^{-1}=N_1, \forall h\in H_1.$ Let $\hat{N_1}$ be the set of
homomorphisms from $N_1$ to $U(1)$ and form a group
$\hat{N_1}\rtimes H_1$. Assume that $H_1$ acts irreducibly on
$\hat{N_1}$ and $H_1$ is solvable. Then the number of cocycles from
$H_1$ to $\hat{N_1},$ i.e., maps $\mu: H_1\rightarrow \hat{N_1}$
such that
$$
\mu(h_1)\mu(h_2)^{h_1}=\mu(h_1h_2)
$$
is less or equal to  $(|N_1|-1)|H_1|.$
\end{lemma}
\prf If $H_1$ acts trivially on $\hat{N_1},$ since $H_1$ acts
irreducibly on $\hat{N_1},$ it follows that $\hat{N_1}$ is an
abelian group of prime order, and the number of cocycles from $H_1$
to $\hat{N_1}$ is bounded by $|H_1|.$ If  $H_1$ acts nontrivially on
$\hat{N_1},$ then $H_1$ is a maximal subgroup of $\hat{N_1}\rtimes
H_1$ with trivial core, and since $H_1$ is solvable, all cocycles
from $H_1$ to $\hat{N_1}$ are coboundaries by Th.16.1 of
\cite{solvable}, and is bounded by $|\hat{N_1}|.$ Since $|H_1|>1$ in
this case the lemma is proved. \qed
\begin{theorem}\label{depth2th}
Let $\A=L(N)\rtimes_\xi H$ be Kac algebras of Izumi-Kosaki type as
in definition \ref{ikalg}. Assume that $N,H$ are solvable groups.
Then the number of maximal (resp. minimal) right coideals is less
than the dimension of $\A.$
\end{theorem}
\prf Assume that $\B$ is a right coideal of $\A$ and let
$(N_1,H_1,\lambda)$ be the triple as in Th. \ref{coideal}. Let us
first prove the minimal case. Since $\B\supset (L(N/N_1),e),$ if
$N_1\neq N,$ we must have $\B= (L(N/N_1),e),$ and $N_1$ must be
maximal in $N.$ The number of such $N_1$ is less than $|N|$ by Th.
\ref{relativeth}.\par If $N_1=N,$ then $H_1\neq e.$ Let $\Z_p\leq
H_1$ be any minimal subgroups of $H_1,$ then the triple
$(N,\Z_p,\lambda)$ will give rise to a right coideal of $\A$ by Th.
\ref{coideal} which is contained in $\B.$ It follows that
$H_1=\Z_p.$ By Lemma \ref{count1}, the number of such triple is
bounded by $|\hat{N}|\leq |N|.$ So minimal right coideal of $\A$ is
bounded by the sum of number of maximal subgroups of N and the
product of the number of minimal subgroups of $H$ by $|N|,$ and it
follows that the number of minimal right coideals is less than the
dimension of $\A.$\par Now assume that $\B$ is maximal. If $N_1$ is
trivial, then $H_1$ is maximal in $H,$ and by Th. \ref{relativeth}
the number of  maximal $H_1$ is less than $|H|.$\par If $N_1$ is
nontrivial, then $\bigoplus_{h\in H_1}(L(N),h)\supset \B,$ and it
follows that $H_1=H.$ We claim that $N_1$ is generated by $\Ad_H
(x)$ for any nontrivial $x\in N_1.$ In fact let $N_1'\subset N_1$ be
a subgroup generated by  $\Ad_H (x)$ for a nontrivial $x\in N_1.$
Then the right coideal determined by the triple $(N_1',H,\lambda)$
contains $\B$, and by maximality of $\B$ we have $N_1'=N_1.$ It
follows that $H$ acts irreducibly on $N_1/[N_1,N_1],$ and therefore
acts irreducibly on its dual $\hat{N_1}.$ By lemma \ref{count2} such
$\B$ with fixed $(N_1,H)$ is bounded by $(|N_1|-1)|H|.$ Note that
different $N_1$'s intersect only at identity. It follows that the
number of maximal $\B$'s is bounded by
$$
(|H|-1)+|H|(|N|-1)=|H||N|-1.
$$
\qed

We consider the intermediate subfactors of $\L^\B\subset L$
corresponding to $\B$ as in Th. \ref{coideal}.
\begin{lemma}\label{jb}
The dimension of second higher relative commutant associated with
the subfactor $\L^{\B}\subset \L$ is given by
$$
\sum_{h\in H_1} \dim(C_R(h)\cap C_L(h))
$$
where
$$
C_R(h)\cap C_L(h):=\{f\in L(N),
R_{n_1,\eta_h}f=\lambda(n_1,h)f=L_{n_1,\eta_h}f\}.
$$
\end{lemma}
\prf This follows from \S3 of \cite{NV} and  Th. \ref{coideal}.\qed
\begin{lemma}\label{jb1}
Let $\B$ be as in Th. \ref{coideal} with triple $(N_1,H_1,\lambda).$
Suppose that $\lambda$ can be extended to $N_i\supset N_1$ such that
the triple $(N_i,H_1,\lambda),i=1,2,...,n$ gives a right coideal of
$\A$ via Th. \ref{coideal}, and $N_i\cap N_j=N_1, \forall i\neq j.$
Let $k_i$ be the number of double cosets of $N_1$ in $N_i.$ Then
$$\dim(C_R(h)\cap C_L(h))\geq 1+(k_2-1)+...+(k_n-1).
$$
\end{lemma}
 \prf On each double coset $N_1bN_1$ of $N_i,$ we can define a
function such that its value on the double coset is simply the value
of $\lambda (.,h)$ and zero elsewhere. It is easy to check that
these functions belong to $C_R(h)\cap C_L(h),$ and they are linearly
independent since they have different support, and the lemma
follows. \qed

The following two lemmas are straightforward consequences of
definitions:
\begin{lemma}\label{db1}
Let $N_2\supset N_1.$ Then the number of homomorphisms from $N_2$ to
$U(1)$ which  takes value $1$ on $N_1$ is bounded by the number of
double cosets of $N_1$ in $N_2.$
\end{lemma}
\begin{lemma}\label{miniext}
Let $N_2\subset N$ be a minimal extension of $N_1$ which is
$\Ad_{H_1}$ invariant. Then the natural action of $H_1$ on
$N_2/N_1[N_2,N_2]$ is irreducible. The dual of $N_2/N_1[N_2,N_2]$ is
abelian group of homomorphisms from $N_2$ to $U(1)$ which  takes
value $1$ on $N_1.$
\end{lemma}
\begin{theorem}\label{relative2}
Let $\B\subset \A$ be a right coideal of $\A$ as in Th.
\ref{coideal}. Then both minimal version and maximal version of
Conjecture \ref{wall} is true for subfactors $\L^\B\subset \L$ when
$H,N$ are solvable.
\end{theorem}
\prf Let $\B_i\subset \B$ and $(N_i,H_i,\lambda_i)$ be the
associated triple of $\B_i$ as in Th. \ref{coideal}. We have
$N_i\supset N_1, H_i\subset H_1,$ and $\lambda_i$ agrees with
$\lambda$ on $N_1\times H_i.$\par Let us first prove the maximal
case. Assume that $\B_i\subset \B,i=1,2,3,..,m$ is the list of
maximal right coideals of $\A$ that is contained in $\B$. Let
$(N_j,H_j), j=1,2,3,...,n$ be the list of different pairs of groups
that are associated with $\B_i's$ as in Th. \ref{coideal}.

If $N_j=N_1,$ then $H_j\subset H_1$ is maximal in $H_1.$ The number
of such maximal $H_j$ is less than $|H_1|$ by Th. \ref{relativeth}.
If $N_j\neq N_1,$ since $\Ad_{H_j}(N_1)=N_1,$ it follows that
$H_j=H_1,$ and $N_j$ is a minimal extension of $N_1$ which is
invariant under $\Ad_{H_1}.$ Let $k_j$ be the number of double
cosets of $N_1$ in $N_j.$ Note that these $N_j$'s only intersect at
$N_1. $ By Lemma \ref{miniext},Lemma \ref{db1} and Lemma
\ref{count2} the number of such $\B_i$ with fixed $N_j$ is bounded
by $(k_j-1)|H_1|.$ So $m\leq |H_1|-1 + [(k_2-1)+...+(k_n-1)]|H_1|$
and the theorem follows from Lemma \ref{jb} and Lemma \ref{jb1}.\par
Now assume that $\B_i\subset \B,i=1,2,3,..,m$ is the list of minimal
right coideals of $\A $ that is contained in $\B$. Let $(N_j,H_j),
j=1,2,3,...,p$ be the list of different pairs of groups that are
associated with $\B_i's$ as in Th. \ref{coideal}. By considering
$(L(N/N_j),e)$ it follows that if $H_j$ is trivial, then $N_j\subset
N$ is a maximal subgroup, and the number of such maximal subgroups
is bounded by the double cosets of $N_1$ in $N$ by Th.
\ref{relativeth}.  If $H_j$ is nontrivial, then $N_j=N,$ and it
follows that $H_j$ has to be a minimal nontrivial subgroup of $H_1.$
The number of such subgroups of $H_1$ is bounded by $|H_1|-1$ if
$H_1$ is not an abelian group of prime order, and $1$ if $H_1$ is an
abelian group of prime order. For each fixed $(N,H_j),$ the possible
$\lambda_j$'s which agrees with $\lambda$ on $N_1\times H_j$ is
clearly bounded by the number of homomorphisms from $N$ to $U(1)$
which vanishes on $N_1$, and by Lemma \ref{db1} this number is
bounded by the number of double cosets of $N_1$ in $N$ which is
denoted by $p_1$.  It follows that $p\leq p_1-1+ (|H_1|-1)p_1,$ and
by Lemma \ref{jb} we are done.\qed
\begin{remark}
If we set $H$ to be a trivial group in Th. \ref{relative2}, then we
recover Th. \ref{relativeth}.
\end{remark}

\begin{remark}
For subfactor $\L^\A\subset \L^\B,$ we can map its intermediate
subfactors to certain right coideals of $\hat{\A}$ which is the dual
of $\A$ (cf. \S2 of \cite{IK}). Essentially similar argument as in
the proof of Th. \ref{relative2} shows that subfactor
 $L^\A\subset L^\B$ verifies the maximal and minimal version of Conjecture \ref{wall} when $H,N$ is
solvable.
\end{remark}

\section{Tensor Product Conjecture}\label{tensorsection}
\begin{lemma}\label{tensorl}
Let $G = X \times Y$ be finite groups with $|X|=x$and $|Y|=y$. Then
the number  of maximal subgroups of G which contain neither $X$ nor
$Y$ is at most $(x-1)(y-1)$ (with equality if and only if X and Y
are elementary abelian 2-groups).
\end{lemma}

\prf  Let $M $ be a maximal subgroup of $G$ containing neither $X$
nor $Y$. Let  $f~:~G\rightarrow G/K$ be the natural homomorphism where
$K$ is the core of $M $ in $G$. Then $f(X)$ and $f(Y)$ are normal
nontrivial subgroups which commute in the primitive group $G/K$ and
moreover, they generate $G/K$ together. By the
Aschbacher-O'Nan-Scott Theorem (although this can be proved easily
in this case) this implies that either $f(X)=f(Y)$ has prime order
$p$ for some prime $p$ or $G/K = f(X) \times f(Y) = S \times S$ with
$S$ a nonabelian simple group.

Thus,  passing to the quotient  by the intersection of all the cores of such
maximal subgroups, we may assume that $X$ and $Y$ are direct products
of simple groups.  Write $X=  \prod_p X_p \times \prod_S X_S$ where $X_p$
is the maximal elementary abelian $p$-quotient of $X$ and $X_S$
is the maximal quotient of $X$ that is a direct product of nonabelian simple groups
each isomorphic to $S$.   Write $Y$ in a similar manner.  The previous paragraph
shows that we may reduce to the case that either $X$ and $Y$ are each elementary
abelian $p$-groups or are both direct products of a fixed nonabelian simple group $S$.

In the first case, it is trivial to see that the total number of maximal subgroups
is $(xy-1)/(p-1)$ while the number of maximal subgroups containing either $X$ or $Y$
is  $(x-1)/(p-1) + (y-1)/(p-1)$.  Thus, the total number of maximal subgroups containing
neither $X$ nor $Y$ is $(x-1)(y-1)/(p-1)$.

In the second case,  write $X=S^a$ and $Y=S^b$.  If $M$ is a maximal subgroup
not containing $X$ or $Y$, then $M \cap X$ is normal in $X$ with
$X/(M \cap X) \cong S$.   Thus $M$ is a direct factor of $X$
isomorphic to $S^{a-1}$.  There are $a$ such factors.    Thus, the number
of maximal subgroups of $X \times Y$ not containing $X$ or $Y$ is $abc$
where $c$ is the number of maximal subgroups of $S \times S$ not containing
either factor.  This is precisely $|\aut(S)|$  (since any such maximal subgroup
is $\{s, \sigma(s) | s \in S\}$ where $\sigma \in \aut(S)$).   Thus, in this case
the number of maximal subgroups not containing either factor is $ab|\aut(S)|$.
To complete the proof, we only need to know that $|\aut(S)| <( |S|-1)^2$.

This is well known (and in fact much better inequalities can be shown).   All such
existing proofs depend upon the classification of finite simple groups.  We note
that the inequality we need follows from the fact that every finite nonabelian
simple group can be generated by two elements  (note that if $s \in S$, then
$|\{\sigma(s)| \sigma \in \aut(S)\} < |S| -3 $ (for there are at least $4$ different
orders of elements and so at the very least $4$ different orbits on $S$  -- if
$s,t$ are generators, then any automorphism is determined by its images
on $s,t$ whence the inequality).
\qed

The following corollary follows immediately:

\begin{corollary}\label{tensorw}
Let $G = X \times  Y$ be finite groups such that both $X$ and $Y$
verify Wall's conjecture, then $G$ also verifies Wall's conjecture.
\end{corollary}
Based on Lemma \ref{tensorl}, we propose the following tensor
product conjecture:
\begin{conjecture}\label{tensorc}
Let $N_i\subset M_i, i=1,2$ be two irreducible subfactors with
finite index. Then the number of minimal intermediate subfactors in
$N_1\bigotimes N_2\subset M_1\bigotimes M_2$ which is not of the
form $N_1\bigotimes P, P\bigotimes N_2$ is less or equal to
$(n_1-1)(n_2-1)$ where $n_i$ is the dimension of second higher
relative commutant of $N_i\subset M_i, i=1,2.$
\end{conjecture}
This conjecture is nontrivial even for subfactors coming from
groups, where we have seen in the proof of Lemma \ref{tensorl} that
we have used classification of finite simple groups to bound the
number of automorphisms of a simple group.

{\footnotesize

config: arxiv