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
Links:
[1] https://www.mpim-bonn.mpg.de/taxonomy/term/39
[2] https://www.mpim-bonn.mpg.de/node/3444
[3] https://www.mpim-bonn.mpg.de/node/3050