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Workshop

\(L^2\)-triviality, Hausdorff dimension and Anosov subgroups

  • Hee Oh (Yale, USA)
E1 05 (Leibniz-Saal)

Abstract

For a discrete subgroup $\Gamma$ of a semisimple Lie group $G$, the space $L^2(\Gamma\backslash G)$ plays a crucial role in bridging representation theory with dynamics. We define $\Gamma$ to be $L^2$-trivial if $L^2(\Gamma\backslash G)$ is weakly equivalent to $L^2(G)$. Determining $L^2$-triviality for a given discrete subgroup is challenging. We present a criterion for Anosov subgroups to be $L^2$ trivial based on the Hausdorff dimension of their limit sets. For example, we obtain that the image of any surface subgroup in a real split higher rank simple Lie group under a positive representation is $L^2$-trivial. We also provide an example of projective Anosov subgroups that are not $L^2$-trivial. It remains an open question whether all Borel Anosov subgroups in higher rank are $L^2$-trivial. This talk is based on several independent joint works with S. Edwards, with M. Fraczyk, and with S. Dey and D. Kim.

Antje Vandenberg (administrative contact)

Max Planck Institute for Mathematics in the Sciences Contact via Mail

J. Audibert, X. Flamm, K. Tsouvalas, T. Weisman (organizational contact)

Olivier Guichard

Université de Strasbourg

Fanny Kassel

Institut des Hautes Études Scientifiques

Anna Wienhard

Max Planck Institute for Mathematics in the Sciences