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

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.