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 $\Gamma$ 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 $\Gamma$ on a quotient of the 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 [5]). In the second part of the paper we study the distribution of $\Gamma$-orbit points in $X$. As a generalization of the critical exponent $\delta (\Gamma )$ of $\Gamma$ we consider for any $\theta \in {\mathbb{R}}_{\ge 0}^r$, $\Vert \theta \Vert =1$, the exponential growth rate ${\delta }_{\theta }(\Gamma )$ of the number of orbit points in $X$ with prescribed „slope" $\theta$. In analogy to Quint's result in [26] we show that the homogeneous extension ${\Psi }_{\Gamma }$ to ${\mathbb{R}}_{\ge 0}^r$ of ${\delta }_{\theta }(\Gamma )$ as a function of $\theta$ is upper semi-continuous, concave and strictly positive in the relative interior of the intersection of the limit cone with the vector subspace of ${\mathbb{R}}^r$ it spans. This shows in particular that there exists a unique slope ${\theta }^{\ast }$ for which ${\delta }_{{\theta }^{\ast }}(\Gamma )$ is maximal and equal to the critical exponent of $\Gamma$.

We notice that an interesting class of product spaces as above comes from the second alternative in the Rank Rigidity Theorem ([(12, Theorem A]) for CAT$(0)$-cube complexes. Given a finite-dimensional CAT$(0)$-cube complex $X$ and a group $\Gamma$ of automorphisms without fixed point in the geometric compactification of $X$, then either $\Gamma$ contains a rank one isometry or there exists a convex $\Gamma$-invariant subcomplex of $X$ which is a product of two unbounded cube subcomplexes; in the latter case one inductively gets a convex $\Gamma$-invariant subcomplex of $X$ which can be decomposed into a finite product of rank one Hadamard spaces.