Rigidity, unitary representations of semisimple groups, and fundamental groups of manifolds with rank one transformation group

Yehuda Shalom

Review from Zentralblatt MATH:

Let $G$ be a simple Lie group locally isomorphic to $SO(n,1)$ or $SU(n,1),$ and let $\Lambda$ be a discrete subgroup of $G.$ The critical exponent $\delta(\Lambda)$ of $\Lambda$ is $$ \delta(\Lambda)=\inf \{s \mid \sum_{\lambda\in \Lambda}e^{-sd(\lambda x_0,x_0)}<\infty\}, $$ where $d$ is a $G$-invariant metric on the associated Riemannian symmetric space $G/K$ and $x_0$ is a fixed origin in $G/K.$ One of the main results of this truly remarkable paper is as follows.

Theorem A. Assume that $G$ is not locally isomorphic to $SL(2, {\Bbb R}),$ and let $\Gamma$ be a lattice in $G.$ If $\Gamma$ is isomorphic to a discrete subgroup $\Lambda$ of some $SO(m,1)$ or $SU(m,1),$ then $\delta(\Gamma)\leq \delta(\Lambda).$ This generalizes a result of {\it C. Yue} [Ann. Math. 143, 331-355 (1996; Zbl 0883.53047)]). The above theorem is extended in order to obtain a rigidity result in the situation where $G$ acts on a compact manifold $M,$ preserving a geometric structure, and $\Lambda$ is the fundamental group of $M.$ One of the most interesting aspects of this paper is the novelty of the approach. It is based on the study of unitary representations of the Lie group $G$, more precisely, on estimates of their matrix coefficients and on their cohomology. The key notion is the following invariant $p(G)$ which makes sense for an arbitrary group: $p(G)$ is the infimum over all real numbers $0\leq p\leq \infty$ for which there exists a unitary representation $(\pi, {\cal H})$ of $G$ with non-vanishing first cohomology group $H^1(G,\pi)$ which is strongly $L^p$( that is, the matrix coefficients of $\pi$ corresponding to a dense subspace of ${\cal H}$ are in $L^{p+\varepsilon}(G)$ for all $\varepsilon>0$). A crucial result in the paper is the following:

Theorem B. With notation and hypotheses as in Theorem A, one has: $p(G)=p(\Gamma)=\delta(\Gamma)=\delta (G)$ (where $\delta (G)=n-1$ if $G=SO(n,1)$ and $\delta(G)=2n$ if $G=SU(n,1)$). The proof is based in an essential way on the following remarkable result.

Theorem C. With notation and hypotheses as in Theorem A, assume moreover that $G$ is not locally isomorphic to $SL(2, {\Bbb C}).$ Let $\pi$ be a unitary representation of $\Gamma.$ Then $H^1(\Gamma,\pi)$ is isomorphic to $H^1(G,\text{Ind}\pi),$ where $\text{Ind}\pi$ is the unitary representation of $G$ induced by $\pi.$ This result is known (and easy to prove) in case $\Gamma$ is cocompact, so that the real (and difficult) issue is when $\Gamma$ is not cocompact. In fact, this result may fail for (non-cocompact) lattices in the groups excluded in the above theorem, that is, $SL(2,{\Bbb R}) $ or $SL(2,{\Bbb C}).$

Reviewed by Mohamed B.Bekka

Keywords: rigidity; lattices; critical exponent; unitary representation

Classification (MSC2000): 22E40 22E46

Full text of the article:

• PDF file (444 kilobytes)