Asymptotic geometry in higher products of rank one Hadamard spaces

Abstract

Given a product X of locally compact rank one Hadamard spaces, we study asymptotic properties of certain discrete isometry groups of X. First we give a detailed description of the structure of the geometric limit set and relate it to the limit cone; moreover, we show that the action of on a quotient of th e regular geometric boundary of X is minimal and proximal. This is completely analogous to the case of Zariski dense discrete subgroups of semi-simple Lie groups acting on the associated symmetric space (compare [Ben97]). In the second part of the paper we study the distribution of -orbit points in X: As a generalization of the critical exponent δ() of we consider for any θ ∈ R r0 , kθk = 1, the exponential growth rate δ�() of the number of orbit points in X with prescribed “slope” θ. In analogy to Quint’s result in [Qui02b] we show that the homogeneous extension � to R r0 of δ�() as a function of θ is upper semi-continuous, concave and strictly positive in the relative interior of the limit cone. This shows in particular that there exists a unique slope θ � for which δ��() is maximal and equal to the critical exponent of . We notice that an interesting class of product spaces as above comes from the second alternative in the Rank Rigidity Theorem ( [CS11, Theorem A]) for CAT(0)-cube complexes: Given a finite-dimensional CAT(0)-cube complex X and a group of automorphisms without fixed point in the geometric comp actification of X, then either contains a rank one isometry or there exists a conve x -invariant subcomplex of X which is a product of two unbounded cube subcomplexes; in the latter case one inductively gets a convex -invariant s ubcomplex of X which can be decomposed into a finite product of rank one Hadamard spaces. So our results imply in particular that classical properties of discrete subgroups of higher rank Lie groups as in [Ben97] and [Qui02b] also hold for certain discrete isometry groups of reducible CAT(0)-cube complexes.