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

Abstract

Abstract Let G be a large group acting on a biregular tree T and $\Gamma \le G$ Γ ≤ G a geometrically finite lattice. In an earlier work, the authors classified orbit closures of the action of the horospherical subgroups on /\Gamma $ G / Γ . 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 /\Gamma $ G / Γ . 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 Følner sequence do not converge. In the last part, as a by-product of our methods, we show that projections to $\Gamma \backslash T$ Γ \ T of the uniform distributions on large spheres in the tree T converge to a natural probability measure on $\Gamma \backslash T$ Γ \ T . Finally, we apply this equidistribution result to a lattice point counting problem to obtain counting asymptotics with exponential error term. Abstract Let G be a large group acting on a biregular tree T and $\Gamma \le G$ Γ ≤ G a geometrically finite lattice. In an earlier work, the authors classified orbit closures of the action of the horospherical subgroups on /\Gamma $ G / Γ . 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 /\Gamma $ G / Γ . 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 Følner sequence do not converge. In the last part, as a by-product of our methods, we show that projections to $\Gamma \backslash T$ Γ \ T of the uniform distributions on large spheres in the tree T converge to a natural probability measure on $\Gamma \backslash T$ Γ \ T . Finally, we apply this equidistribution result to a lattice point counting problem to obtain counting asymptotics with exponential error term.