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

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.