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

TL;DR

This paper studies the behavior of horospherical actions on trees, proving equidistribution and no escape of mass for geometrically finite lattices, while revealing new phenomena for non-geometrically finite lattices, with applications to lattice point counting.

Contribution

It establishes equidistribution and absence of escape of mass for horospherical actions on trees with geometrically finite lattices, and introduces new dynamical phenomena for non-geometrically finite cases.

Findings

Dense orbits equidistribute to Haar measure.

Escape of mass occurs in non-geometrically finite cases.

Projections of large sphere distributions converge to a natural measure.

Abstract

Let $G$ be a large group acting on a biregular tree $T$ and $\Gamma \leq G$ a geometrically finite lattice. In an earlier work, the authors classified orbit closures of the action of the horospherical subgroups on $G/\Gamma$. In this article we show that there is no escape of mass and use this to prove that, in fact, dense orbits equidistribute to the Haar measure on $G/\Gamma$. On the other hand, we show that new dynamical phenomena for horospherical actions appear on quotients by non-geometrically finite lattices: we give examples of non-geometrically finite lattices where an escape of mass phenomenon occurs and where the orbital averages along a Folner sequence do not converge. In the last part, as a by-product of our methods, we show that projections to $\Gamma \backslash T$ of the uniform distributions on large spheres in the tree $T$ converge to a natural probability measure on $\Gamma \backslash T$. Finally, we apply this equidistribution result to a lattice point counting problem to obtain counting asymptotics with exponential error term.