Finite-sided Dirichlet domains and Anosov subgroups

We consider Dirichlet domains for Anosov subgroups Γ of semisimple Lie groups G acting on the associated symmetric space $G/K$. More precisely, we consider certain Finsler metrics on $G/K$ and a sufficient condition so that every Dirichlet domain for Γ is finite-sided in a strong sense. Under the same condition, the group Γ admits a domain of discontinuity in a flag manifold where the Dirichlet domain extends to a compact fundamental domain. As an application we show that Dirichlet-Selberg domains for n-Anosov subgroups Γ of SL(2n,R) are finite-sided when all singular values of elements of Γ diverge exponentially in the word length. For every d≥ 3, there are projective Anosov subgroups of SL(d,R) which do not satisfy this property and have Dirichlet-Selberg domains with infinitely many sides. More generally, we give a sufficient condition for a subgroup of SL(d,R) to admit a Dirichlet-Selberg domain whose intersection with an invariant convex set is finite-sided.