Dirichlet domains for Anosov subgroups

Abstract

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