Dirichlet domains for Anosov subgroups

Abstract

We introduce a sufficient condition for a finitely generated subgroup $\Gamma$ of a semisimple Lie group $G$ to admit finite-sided Dirichlet domains for polyhedral Finsler metrics on the symmetric space $G/K$. The condition always implies the $\Theta$-Anosov condition for some $\Theta$, and can be arranged to be equivalent to the $\Theta$-Anosov condition when $G$ is simple and $\Theta$ is the set of long roots or the set of short roots. The Dirichlet domain we obtain extends to a fundamental domain for the action of $\Gamma$ on a domain of discontinuity in a flag manifold. For instance, Borel Anosov subgroups of $\mathrm{SL}(d,\mathbb{R})$ have finite-sided Dirichlet domains for the Hilbert metric on the symmetric space which extends to the space of line-hyperplane flags, and $n$-Anosov subgroups of $\mathrm{Sp}(2n,\mathbb{R})$ have finite-sided Dirichlet-Selberg domains in $\mathrm{SL}(2n,\mathbb{R})/\mathrm{SO}(2n)$ which extend to a domain in projective space bounded by quadrics.