Measure Rigidity for Horospherical Subgroups of Groups Acting on Trees

Abstract

Abstract We prove analogues of some of the classical results in homogeneous dynamics in nonlinear setting. Let $ be a closed subgroup of the group of automorphisms of a biregular tree and $\Gamma \leq G$ a discrete subgroup. For a large class of groups $, we give a classification of the probability measures on /\Gamma $ invariant under horospherical subgroups. When $\Gamma $ is a cocompact lattice, we show the unique ergodicity of the horospherical action. We prove Hedlund’s theorem for geometrically finite quotients. Finally, we show equidistribution of large compact orbits.

Abstract We prove analogues of some of the classical results in homogeneous dynamics in nonlinear setting. Let $ be a closed subgroup of the group of automorphisms of a biregular tree and $\Gamma \leq G$ a discrete subgroup. For a large class of groups $, we give a classification of the probability measures on /\Gamma $ invariant under horospherical subgroups. When $\Gamma $ is a cocompact lattice, we show the unique ergodicity of the horospherical action. We prove Hedlund’s theorem for geometrically finite quotients. Finally, we show equidistribution of large compact orbits.