Measure Rigidity for Horospherical Subgroups of Groups Acting on Trees

Abstract

We prove analogues of some of the classical results in homogeneous dynamics in nonlinear setting. Let $G$ 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 $G$, we give a classification of the probability measures on $G/\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.

(Non)-escape of mass and equidistribution for horospherical actions on trees

“…In our geometric setup, the role of Ad-unipotent subgroups in classical homogeneous dynamics is played by the horospherical subgroups G 0 η of G, for η ∈ ∂ T . In the earlier work [13], the authors classified Borel probability measures on G/ invariant under G 0 η -action for large class of groups G and general lattices , establishing an analogue of Dani's result in [15]. Moreover, it was shown that when is geometrically finite, G 0 η -orbits are either compact or dense, as in the classical result of Hedlund [30] on the horocycle flow on finite volume hyperbolic surfaces.…”

“…This can be done using the Howe-Moore property, established in our setting in [11] and amenable ergodic theorem [34]. Our topological result in [13] says, however, that all points x ∈ X that do not lie in a compact G 0 η -orbit have dense orbits. Therefore, the immediate question arises whether every dense orbit equidistributes to the Haar measure on G/ .…”

“…Then for any M-invariant compact subset O with non-empty interior in G 0 η , the sequence (O t := a t Oa −t ) t∈N constitutes a Følner sequence in G 0 η (see e.g. [13,Lemma 2.10]). In the sequel, we shall refer to such sequences O t as good Følner sequences.…”

“…In the proof of Theorem A, exploiting the underlying geometric setting, we translate the problem of understanding the distribution of G 0 η -orbit in G/ to the language of Markov chains, where it appears as a problem of controlling the distributions of a Markov chain with changing starting distributions. We then rely on two main ingredients: the first is a qualitative description of the behaviour of the discrete geodesic flow on G/ , as studied in [13]. This allows us to understand the behaviour of starting distributions of the Markov chain.…”

“…We remark that for uniform lattices, one can use Margulis' orbit thickening argument to show that U -action is uniquely ergodic (see Mohammadi [38], Ellis-Perrizo [22] or [13,Lemma 6.3]). It is also worth noting that for non-uniform quotients, using our geometric approach, one can show the version of the previous corollary for the U -action (instead of MU ).…”