Dirichlet domains for Anosov subgroups

Dirichlet domains provide polyhedral fundamental domains for discrete subgroups of the isometries of the hyperbolic space. Selberg introduced a similar construction of a polyhedral fundamental domain for the action
of discrete subgroups of the higher rank Lie group $\SL(n,\R)$ on the projective model of the associated symmetric space.  His motivation was to study uniform lattices. We will address the following question asked
by Kapovich: for which Anosov subgroups are these domains finite-sided ?

We will first consider an example of a Anosov subgroup for which this domain can have infinitely many sides. We then provide a sufficient condition on a subgroup to ensure that the domain is finitely sided in a strong sense. Finally we will mention some generalizations of this result to Dirichlet domain in other symmetric spaces for some particular Finsler metrics. This is joint work with Max Riestenberg