Measure rigidity for horospherical subgroups of groups acting on trees

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