The Cheeger constant of simply connected, solvable Lie groups
Norbert Peyerimhoff
Evangelia Samiou
math0206055.tex
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\begin{document}

\title{The Cheeger constant of simply connected, solvable Lie groups}
\author{Norbert Peyerimhoff, Evangelia Samiou}
\date{}
\maketitle

\parbox{12cm}{
\small{\bf Abstract:}
We show that the Cheeger isoperimetric constant of a solvable simply
connected Lie group $G$ with Lie algebra $\G$ is
$h(G)=\max_{H\in\G,||H||=1} \tr(\ad (H))$.
}

\section{Introduction}
The Cheeger isoperimetric constant $h(M)$ of a complete noncompact
Riemannian manifold $M$ is defined by $$h(M)= \inf_{K\subset M}
\frac{\area (\partial K)}{\vol (K)},$$ where $K$ ranges over all
connected, open submanifolds of $M$ with compact closure and smooth
boundary.  For the bottom of the spectrum $\lambda(\Omega)$, there is
Cheeger's inequality \cite{c}, \cite {bpp}, \cite{cha}
$$\lambda(\Omega) \geq
\frac{h(\Omega)^2}{4}.$$
There is a converse of Cheeger's inequality due to Peter Buser,
\cite{bu},
\cite{bpp}: There exist constants $c_{1}$ and $c_{2}$, depending on a
lower bound on the Ricci curvature of $M$ such that
$$ \lambda\leq c_{1}h+c_{2}h^{2}.$$
Further relations between isometric and spectral
properties (in particular, estimates of the heat kernel) can be found
in the work of, e.g.,
%xxx
Varopoulos, Coulhon, Saloff-Coste, Grigor'yan \cite{gr}
and Pittet \cite{pi}. For connections between the Cheeger constant and
Kazhdan's property T see, e.g., \cite{Br} or \cite{Leu}.

%Cheeger constants are also useful in the theory of foliations. It was
%shown
%by Plante (\cite{pl}, \cite{su}) that if $M$ is a foliated Riemannian
%manifold
%with a leaf $\mathcal{F}\subset M$ satisfying $h(\mathcal{F})=0$ then
%the
%foliation admits an invariant transversal measure.

If $h(M)> 0$ one can easily see that $M$ has exponential volume
growth. The converse is not true. Hoke \cite{h} has shown that
$h(G)=0$ for a simply connected Lie group with left invariant metric
if and only if $G$ is unimodular and amenable. These Lie groups,
however, have exponential volume growth if they are not of type R, see
\cite{pa}. Examples of this class are horospheres in symmetric spaces
of noncompact type and higher rank orthogonal to the barycenter of a
Weyl chamber, see \cite{p}.

In this note we calculate the Cheeger constant for simply connected
solvable Lie groups.

\section{ The Cheeger constant}
We consider the following situation: Let $G=G_{0}\semidirect\R$ be a
semidirect product with Lie algebra
$\G = \G_{0} \oplus \R$ endowed with a left invariant metric such that
$\G_{0}$
is orthogonal to $\R$.
We denote the unit vector in $\R$ by $H_{0}$ and obtain
a diffeomorphism $\phi: G_{0} \times \R \to G$, $\phi(g_{0},t)=
g_{0} \exp(tH_{0})$.

\begin{lm}\label{mass}
The left invariant Haar-measure $\mu$ on $G$ is given by
\begin{equation}\label{measure}
d\mu(g_{0}, t) = \det(e^{\ad(-tH_{0})}) d\nu(g_{0})dt
= e^{-t\,\tr(\ad(H_{0}))} d\nu(g_{0})dt     \ ,
\end{equation}
where $d\nu$ is the left invariant Haar-measure on
$G_{0}=G_{0}\times\{0\}\subset G$.
\end{lm}
\noindent

Proof: Note that $d\nu dt$ is left $G_0$-invariant and right
$\R$-invariant.
Consequently, the left invariant Haar-measure on $G$ with respect to the

above
diffeomorphismus
is given by $d\mu(g_{0},t)=\delta(t)d\nu(g_{0})dt$. We will calculate
$\delta(t)$.
Let $a=\exp(sH_{0})$ and $f\in C^\infty_0(G) $. Then
\begin{eqnarray*}
\int_{G} f(ag)d\mu(g) &=& \int_{\R} \int_{G_{0}}
f(ag_{0}\exp(tH_{0}))\delta(t)d\nu(g_{0})dt  \\
                      &=& \int_{\R} \int_{G_{0}}
f(\psi(g_{0})a\exp(tH_{0}))\delta(t)d\nu(g_{0})dt,
\end{eqnarray*}
where $\psi:G_{0}\to G_{0}, \psi(g_{0})=ag_{0}a^{-1}$. From the
transformation formula
this becomes
$$ \int_{\R}\delta(t)\int_{G_{0}}f(g_{0} \exp((s+t)H_{0}))|\det
D\psi^{-1}(g_{0})|d\nu(g_0) dt\ .$$
Let $X_{1}, \cdots, X_{n}$ be an orthonormal basis of $\G_{0}$ with
respect to the left invariant metric.
Then the $Y_{j}=\ddt g_{0} \exp(tX_{j})$, $j=1,\ldots , n$ form an
orthonomal basis
of $T_{g_{0}}G_{0}$ and
\begin{eqnarray*}
D\psi^{-1}(g_{0})(Y_{j})&=&\ddt\psi^{-1}(g_{0}\exp(tX_{j}))
=\ddt a^{-1}g_0\exp(tX_j)a\\
&=& dL_{a^{-1}g_0 a}(e^{\ad(-sH_0)}X_j),
\end{eqnarray*}
so we conclude that $|\det(D\psi^{-1}(g_{0}))|=\det(e^{\ad(-sH_0)})$.
By left invariance of $\mu$ we get
\begin{eqnarray*} \int_{G} f(ag)d\mu(g) &=&
 \int_{\R}\delta(t)\det(e^{\ad(-sH_0)})
\int_{G_{0}}f(g_{0} \exp((s+t)H_{0}))d\nu(g_0) dt\ = \\
\int_{G} f(g)d\mu(g) &=&
\int_{\R}\delta(s+t) \int_{G_{0}} f(g_{0} \exp((s+t)H_{0}))d\nu(g_{0})dt
\ ,
\end{eqnarray*}
hence $\delta(s)= \det(e^{\ad(-sH_0)}).$
\proofend

The following theorem contains the main result of this section.
\begin{thm}\label{main}
For the Cheeger constant of a Lie group $G$ as above we have
\begin{equation} \label{mainest}
h(G) \geq |\tr(\ad(H_{0}))|
\end{equation}
Moreover, if $h(G_0) = 0$ then formula \eqref{mainest} holds with
equality.
\end{thm}

Proof: We identify $G$ with $G_0\times\R$ via the diffeomorphism
$\phi$ and denote by $\pi: G\to G_{0}$ the projection. Without loss of
generality we may assume that $\tr(\ad(H_0)) \ge 0$. For a connected,
compact subset $K\subset G$ with smooth boundary $\partial K$ and non
empty interior let $U=\pi(K)\setminus\{\mbox{critical values of }
\pi|_{\partial K}\}$.

On $U$ we define functions $\delta^\pm$ by
$\delta^+(u)=\max\{t\mid (u,t)\in K\}$
and $\delta^-(u)=\min\{t\mid (u,t)\in K\}$, $u\in U$.
The functions $\delta^\pm $ are smooth and by Sard's
theorem the set $U$ has full measure in $\pi(K)$. We denote by
$ \partial K^{\pm}=\{(u,\delta^{\pm}(u))| u\in U \} $
the graphs of $\delta^\pm$.
We estimate the volumes of $K$ and $\partial K$.
>From \eqref{measure} we get
\begin{eqnarray*}
\vol (K) & \leq & \int_{U}\int_{\delta^{-}(u)}^{\delta^{+}(u)}
e^{-t \,\tr(\ad( H_{0}))}dt d\nu(u)               \\
 & \leq & \frac{1}{\tr(\ad(H_{0}))}\int_{U} \(
e^{-\delta^{+}(u)\tr(\ad(H_{0}))}+
e^{-\delta^{-}(u)\tr(\ad(H_{0}))} \) \, d\nu(u) \ .
\end{eqnarray*}
Clearly $\area (\partial K)\geq \area (\partial K^+)+ \area (\partial
K^-)$.
We have
$$\area (\partial K^+)=
\int_U \sqrt{\det
(\langle
D\varphi(u)e_i,D\varphi(u)e_j\rangle_{(u,\delta^+(u))})_{i,j=1\ldots
n-1} }
\ d\nu(u),$$
where $\varphi(u)=(u,\delta^{+}(u))$ and $e_1,\ldots ,e_{n-1}$ is an
orthonormal basis of $T_uG_0$. We estimate the integrand
\begin{eqnarray}
\sqrt{\det(\langle
D\varphi(u)e_i,D\varphi(u)e_j\rangle_{(u,\delta^+(u))}) }
&=& \sqrt{\det(
\langle e_i,e_j\rangle_{(u,\delta^+(u))}+ vv^{\top}) } \nonumber \\
&\geq &\sqrt{\det\(\langle e_i,e_j\rangle_{(u,\delta^+(u))}\) }
\nonumber\\
&=&e^{-\delta^+(u)\tr(\ad(H_{0}))}, \label{norbi}
\end{eqnarray}
where $v^{\top}=(e_{1}(\delta^+),\cdots, e_{n-1}(\delta^+))$ and the
$e_i$'s
on the right hand side are considered as elements in
$T_{(u,\delta^{+}(u))}G$.
%xxx
The equality \eqref{norbi} follows from \eqref{measure}. This together
with the analogous estimate for $\partial K^-$ yields
\begin{equation} \label{form1}
\area (\partial K) \geq \tr(\ad(H_{0})) \vol (K) \ .
\end{equation}
This proves inequality \eqref{mainest}.

Finally, we prove equality of \eqref{mainest} in the case $h(G_0) =
0$. As before, we use the diffeomorphism $G \cong G_0\times \R$.
Let $K_0\subset G_0$ be arbitrary and consider the set
$K=K_0\times [a,b] \subset G$ with boundary
$\partial K= (K_0\times\{a,b\}) \cup (\partial K_0\times [a,b])$.
Direct calculations yield
\begin{equation}\label{volK}
\vol (K) = \vol (K_0) \frac{e^{- a \tr \ad H_0 }-e^{- b \tr \ad H_0 }}
{\tr\ad H_0} \ ,
\end{equation}
as well as
\[ \area(K_0\times\{a,b\}) = \vol(K_0) \left( e^{- a \tr \ad H_0} +
e^{- b \tr \ad H_0 } \right). \]
Given $\epsilon > 0$ we can arrange for
\begin{equation}\label{dk}
\frac{\area(K_0\times\{ a,b\})}{\vol (K)}\leq \tr \ad H_0 + \epsilon
\end{equation}
by choosing $b$ sufficiently large.
Note that \eqref{volK} and the estimate \eqref{dk} also make
sense if $\tr\ad H_0 =0 $.

Let $n = \dim \G_0$. Choosing an orthonormal basis $e_1,\ldots,
e_{n-1}$ of $T_u \partial K_0 \subset \G_0$ and abbreviating $e^{-t
(\ad H_0 )}=A_t$ we finally compute
$$\area(\partial K_0\times [a,b])=\int_a^b
\vol\nolimits_{n-1}(\partial K_0\times \{ t \})  \ dt$$
where the $(n-1)$-dimensional volume is given by
\[ \vol\nolimits_{n-1}(\partial K_0\times {t}) =
\int_{\partial K_0}\sqrt{\det\left(
\langle A_t e_p ,A_t e_q\rangle_{\G_0}\right)_{p,q=1\ldots n-1} }
\ d\vol\nolimits_{\partial K_0}(u).
\]
The integrand can be estimated from above by
$$M:=\max\left\{\left. \sqrt{\det\left(\left. P_U {A_t}^*A_{t}
\right|_{U}\right)} \ \right| \  a\leq t\leq b, \   U\subset \G_0, \
\dim U=n-1 \right\}, $$
where $P_U: \G_0 \to U$ denotes orthogonal projection. Note that $M$ is
independent of the specific choice of $K_0$. Therefore choosing $K_0$
such
that $\vol_{n-1}\partial (K_0)/\vol (K_0) $ is sufficiently small we may

assume that
\begin{equation} \label{form2}
\frac{\area(\partial K_0\times [a,b])}{\vol (K)}\leq \frac{M (\tr \ad
H_0)
(b-a)}{ e^{- a\tr \ad H_0 }-e^{- b\tr \ad H_0}}
\frac{\vol_{n-1}(\partial K_0)}{\vol (K_0)} \leq \epsilon.
\end{equation}
Putting \eqref{dk} and \eqref{form2} together yields $h(G) = \tr \ad
H_0$, since $\epsilon > 0$ was chosen arbitrarily small.
\proofend

Now we consider the particular case of a simply connected solvable Lie
group
$G$ with Lie algebra $\G$ and $\N=[\G, \G]$. In this case we obtain
\begin{coro}\label{cheegformel}
The Cheeger constant of a simply connected solvable Lie group $G$ with
Lie algebra $\G$ is
\begin{equation} \label{mainformula}
 h(G) = \max_{H\in \G, ||H||=1} \tr(\ad(H)) \ .
\end{equation}
\end{coro}
Proof: Let $\G_0$ denote the kernel of the $1$-form
$\alpha\colon\G\to\R$, $X \mapsto \tr\ad X $. Clearly $\G_0$ contains
$\N=[\G, \G]$, since $\N$ is nilpotent. Hence $\G_0$ is an ideal and the

corresponding unimodular Lie group satisfies $h(G_0) = 0$ by Theorem
\ref{main}, which proves the corollary in the case $\G_0 =
\G$. Otherwise $\G_0$ has codimension $1$. Let $H_0$ be the maximum of
$\alpha$ on the unit sphere of $\G$, i.e.
$$\max_{H\in \G, ||H||=1} \tr(\ad(H)) = \tr(\ad(H_0)) \ . $$
Then $H_0 \perp \G_0$ and, again, Theorem \ref{main} implies the
statement
of the Corollary.
\proofend

Remarks: Note that simply connected, solvable Lie groups with left
invariant metric may have curvature of both signs (e.g. horospheres in
symmetric spaces). Corollary \ref{cheegformel} in the particular case
of simply connected strictly negatively curved homogeneous spaces
(NCHS) was proved by Connell \cite{co}. He also showed for NCHS that the

Cheeger constant coincides with the exponential volume growth rate.

Formula \eqref{mainformula} also gives a lower bound for the topological
entropy
of compact Riemannian manifolds whose universal covering is a solvable
Lie group with left invariant metric. This follows from
the lower estimate of the topological entropy by the exponential
volume growth rate proved in \cite{ma}. In the case of a locally
symmetric space $M$ the topological entropy was calculated in
\cite{sp} and agrees with both the exponential volume growth rate and
the Cheeger constant of the universal covering $\tilde M$. It is given
by $\Vert \rho \Vert$, where $\rho$ is the sum of the positive roots
with multiplicities. More refined volume growth calculations in
symmetric spaces were carried out in \cite{kn}.

%It would be interesting to investigate the exponential volume growth
%rate of arbitrary solvable groups.

\bigskip

\noindent {\em Acknowledgements } We are grateful to P. Pansu for
bringing to our attention the paper of H. Hoke. We thank the University
of Cyprus for financial support.

\begin{thebibliography}{99}

\bibitem[Br]{Br} R. Brooks, The spectral geometry of a tower of
coverings,
J. Diff. Geom. 23, 97-107, 1986.

\bibitem[BPP]{bpp} R. Brooks, P. Perry, P. Petersen, On Cheegers
inequality,
Comment. Math. Helvetici 68, 599-621, 1993.

\bibitem[Bu]{bu} P. Buser, A note on the isoperimetric constant,
Ann. Sci. Ec. Norm. Sup. 15, 213-230, 1982.

\bibitem[Cha]{cha} I. Chavel,
Eigenvalues in Riemannian geometry,
Academic Press, 1984.

\bibitem[Che]{c} J. Cheeger,
A lower bound for the smallest eigenvalue of the Laplacian,
"Problems in Analysis", 195-199, Princeton Univ. Press, 1970.

\bibitem[Co]{co} Ch. Connell,
Asymptotic harmonicity of negatively curved homogeneous spaces
and their measures at infinity,
Comm. Anal. Geom., vol. 8, no. 3, 575-633, 2000.

\bibitem[Gr]{gr} A. Grigor'yan,
Estimates of heat kernels on Riemannian manifolds,
"Spectral theory and geometry", 140-225, Cambridge Univ. Press, 1999.

\bibitem[Ho]{h} H. F. Hoke III, Lie groups that are closed at infinity,
Transactions of the AMS 313,Number 2, 721-735, 1989.

\bibitem[Kn]{kn} G. Knieper, On the asymptotic geometry of nonpositively

curved manifolds,
Geom. funct. anal. 7, 755-782, 1997.

\bibitem[Leu]{Leu} E. Leuzinger, Kazhdan's property (T), $L^2$-spectrum
and isoperimetric inequalities for locally symmetric spaces, preprint,
2000.

\bibitem[Ma]{ma} A. Manning, Topological entropy for geodesic flows,
Ann. Math. 110, 567-573, 1979.

\bibitem[Pa]{pa} A. Paterson, Amenability, Math. Surveys and
Monographs, no. 29, AMS, 1988.

\bibitem[Pe]{p} N. Peyerimhoff, Isoperimetric and ergodic properties of
horospheres in symmetric spaces, in A. Katok, R. de la Llave, Y. Pesin,
H. Weiss, editors, Smooth ergodic theory and its applications, Proc.
Symp.
Pure Math., AMS, 2001.

\bibitem[Pi]{pi} Ch. Pittet, The isoperimetric profile of homogeneous
Riemannian manifolds, J. Diff. Geom. 54, 255-302, 2000.

%\bibitem[Pl]{pl} J. F. Plante, Foliations with measure preserving
%holonomy, Ann. Math. 102, 327-361, 1975.

\bibitem[Sp]{sp} R. J. Spatzier, Dynamical properties of algebraic
systems,
Dissertation, Warwick and Maryland 1983.

%\bibitem[Su]{su} D. Sullivan, Cycles for the dynamical study of
%foliated
%manifolds and complex manifolds,
%Inv. Math. 36, 225-255, 1976.


\end{thebibliography}

\noindent
N. Peyerimhoff\\
Mathematische Fakult\"at\\
Ruhr-Universit\"at Bochum\\
Universit\"atsstr. 150\\
44780 Bochum\\
Germany\\
e-mail: peyerim@math.ruhr-uni-bochum.de

\bigskip

\noindent
E. Samiou\\
Department of Mathematics and Statistics\\
University of Cyprus\\
P.O. Box 20537\\
1678 Nicosia\\
Cyprus\\
e-mail: samiou@ucy.ac.cy


\end{document}


math0206055.extracted_bib
\begin{thebibliography}{99}

\bibitem[Br]{Br} R. Brooks, The spectral geometry of a tower of
coverings,
J. Diff. Geom. 23, 97-107, 1986.

\bibitem[BPP]{bpp} R. Brooks, P. Perry, P. Petersen, On Cheegers
inequality,
Comment. Math. Helvetici 68, 599-621, 1993.

\bibitem[Bu]{bu} P. Buser, A note on the isoperimetric constant,
Ann. Sci. Ec. Norm. Sup. 15, 213-230, 1982.

\bibitem[Cha]{cha} I. Chavel,
Eigenvalues in Riemannian geometry,
Academic Press, 1984.

\bibitem[Che]{c} J. Cheeger,
A lower bound for the smallest eigenvalue of the Laplacian,
"Problems in Analysis", 195-199, Princeton Univ. Press, 1970.

\bibitem[Co]{co} Ch. Connell,
Asymptotic harmonicity of negatively curved homogeneous spaces
and their measures at infinity,
Comm. Anal. Geom., vol. 8, no. 3, 575-633, 2000.

\bibitem[Gr]{gr} A. Grigor'yan,
Estimates of heat kernels on Riemannian manifolds,
"Spectral theory and geometry", 140-225, Cambridge Univ. Press, 1999.

\bibitem[Ho]{h} H. F. Hoke III, Lie groups that are closed at infinity,
Transactions of the AMS 313,Number 2, 721-735, 1989.

\bibitem[Kn]{kn} G. Knieper, On the asymptotic geometry of nonpositively

curved manifolds,
Geom. funct. anal. 7, 755-782, 1997.

\bibitem[Leu]{Leu} E. Leuzinger, Kazhdan's property (T), $L^2$-spectrum
and isoperimetric inequalities for locally symmetric spaces, preprint,
2000.

\bibitem[Ma]{ma} A. Manning, Topological entropy for geodesic flows,
Ann. Math. 110, 567-573, 1979.

\bibitem[Pa]{pa} A. Paterson, Amenability, Math. Surveys and
Monographs, no. 29, AMS, 1988.

\bibitem[Pe]{p} N. Peyerimhoff, Isoperimetric and ergodic properties of
horospheres in symmetric spaces, in A. Katok, R. de la Llave, Y. Pesin,
H. Weiss, editors, Smooth ergodic theory and its applications, Proc.
Symp.
Pure Math., AMS, 2001.

\bibitem[Pi]{pi} Ch. Pittet, The isoperimetric profile of homogeneous
Riemannian manifolds, J. Diff. Geom. 54, 255-302, 2000.

%\bibitem[Pl]{pl} J. F. Plante, Foliations with measure preserving
%holonomy, Ann. Math. 102, 327-361, 1975.

\bibitem[Sp]{sp} R. J. Spatzier, Dynamical properties of algebraic
systems,
Dissertation, Warwick and Maryland 1983.

%\bibitem[Su]{su} D. Sullivan, Cycles for the dynamical study of
%foliated
%manifolds and complex manifolds,
%Inv. Math. 36, 225-255, 1976.


\end{thebibliography}