Abstract
Let G be a large group acting on a biregular tree T and Gamma le 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 Følner 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.
Highlights
Let T be a (d1, d2)-biregular tree with d1, d2 ≥ 3
This parallels the classical setting of homogeneous dynamics, where one studies the actions of certain subgroups on a quotient of a linear algebraic group by a lattice
We mention the work of Vatsal [56] in which the equidistribution results of Ratner [47,48] for unipotent dynamics in the p-adic case were applicable with a geometric approach similar to ours
Summary
Let T be a (d1, d2)-biregular tree with d1, d2 ≥ 3. Let G be a non-compact, closed subgroup of Aut(T ). This parallels the classical setting of homogeneous dynamics, where one studies the actions of certain subgroups on a quotient of a linear algebraic group by a lattice. These two worlds intersect, for example, when G = SL2(k), where k is a non-archimedean local field, in which case G naturally acts on the associated Bruhat–Tits tree. We recall that when G is linear, by works of Raghunathan and Lubotzky [35,43], any lattice therein is geometrically finite. It was shown that when is geometrically finite, G0η-orbits are either compact or dense, as in the classical result of Hedlund [30] on the horocycle flow on finite volume hyperbolic surfaces
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