Asymptotic geometry in higher products of rank one Hadamard spaces

  • Gabriele Link

    Karlsruhe Institute of Technology, Germany


Given a product XX of locally compact rank one Hadamard spaces, we study asymptotic properties of certain discrete isometry groups Γ\Gamma of XX. 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 Γ\Gamma on a quotient of the regular geometric boundary of XX 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 Γ\Gamma-orbit points in XX. As a generalization of the critical exponent δ(Γ)\delta(\Gamma) of Γ\Gamma we consider for any θR0r\theta \in \mathbb R_{\ge 0}^r, θ=1\Vert \theta \Vert = 1, the exponential growth rate δθ(Γ)\delta_\theta(\Gamma) of the number of orbit points in XX with prescribed „slope" θ\theta. In analogy to Quint's result in [26] we show that the homogeneous extension ΨΓ\Psi_\Gamma to R0r\mathbb R_{\ge 0}^r of δθ(Γ)\delta_\theta(\Gamma) as a function of θ\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 Rr\mathbb R^r it spans. This shows in particular that there exists a unique slope θ\theta^* for which δθ(Γ)\delta_{\theta^*}(\Gamma) is maximal and equal to the critical exponent of Γ\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)(0)-cube complexes. Given a finite-dimensional CAT(0)(0)-cube complex XX and a group Γ\Gamma of automorphisms without fixed point in the geometric compactification of XX, then either Γ\Gamma contains a rank one isometry or there exists a convex Γ\Gamma-invariant subcomplex of XX which is a product of two unbounded cube subcomplexes; in the latter case one inductively gets a convex Γ\Gamma-invariant subcomplex of XX which can be decomposed into a finite product of rank one Hadamard spaces.

Cite this article

Gabriele Link, Asymptotic geometry in higher products of rank one Hadamard spaces. Groups Geom. Dyn. 10 (2016), no. 3, pp. 885–931

DOI 10.4171/GGD/370