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# Groups, Geometry, and Dynamics

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**Volume 10, Issue 3, 2016, pp. 885–931**

**DOI: 10.4171/GGD/370**

Published online: 2016-09-16

Asymptotic geometry in higher products of rank one Hadamard spaces

Gabriele Link^{[1]}(1) Karlsruhe Institute of Technology, Germany

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 â€žslope" $\theta$. 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 $\theta^*$ for which $\delta_{\theta^*}(\Gamma)$ 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.

*Keywords: *CAT(0)-spaces, products, cubical complexes, discrete groups, rank one isometries, limit set, limit cone, critical exponent

Link Gabriele: Asymptotic geometry in higher products of rank one Hadamard spaces. *Groups Geom. Dyn.* 10 (2016), 885-931. doi: 10.4171/GGD/370