Dirichlet domains for Anosov subgroups

We introduce a sufficient condition for a finitely generated subgroup $\mathrm{\Gamma }$ of a semisimple Lie group $G$ to admit finite-sided Dirichlet domains for polyhedral Finsler metrics on the symmetric space $G\mathrm{/}K$. The condition always implies the $\mathrm{\Theta }$-Anosov condition for some $\mathrm{\Theta }$, and can be arranged to be equivalent to the $\mathrm{\Theta }$-Anosov condition when $G$ is simple and $\mathrm{\Theta }$ 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 $\mathrm{\Gamma }$ on a domain of discontinuity in a flag manifold. For instance, Borel Anosov subgroups of $\mathrm{S}\mathrm{L}(d,\mathbb{R})$ 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 $\mathrm{S}\mathrm{p}(2n,\mathbb{R})$ have finite-sided Dirichlet-Selberg domains in $\mathrm{S}\mathrm{L}(2n,\mathbb{R})\mathrm{/}\mathrm{S}\mathrm{O}(2n)$ which extend to a domain in projective space bounded by quadrics.