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Re­search sem­in­ar "Geo­­met­ric and Har­­mon­ic Ana­lys­is": Colin Dav­alo (Uni Heidel­berg)

Ort: D2.314

Titel: Finite-sided Dirichlet domains for Anosov representations

Abstract: Dirichlet domains provide polyhedral fundamental domains in hyperbolic space for discrete subgroups of the isometry group. 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, for which these domains are finite-sided. However these domains can also be studied for smaller subgroups, and we will consider the following question asked by Kapovich: for which Anosov subgroups are these domains finite-sided ?

Anosov subgroups are hyperbolic discrete subgroups satisfying strong dynamical properties, but are not lattices in higher rank. We will first consider an example of an Anosov subgroup for which the fundamental domain constructed by Selberg has infinitely many sides. We then provide a sufficient condition on a subgroup to ensure that the domain is finitely sided in a strong sense. This is joint work with Max Riestenberg.

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