Asymptotic geometry in higher products of rank one Hadamard spaces

Gabriele Link¹

¹Karlsruhe Institute of Technology, Germany

Groups, geometry, and dynamics, Tome 10 (2016) no. 3, pp. 885-931

Résumé

Given a product X of locally compact rank one Hadamard spaces, we study asymptotic properties of certain discrete isometry groups Γ 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 Γ 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 Γ-orbit points in X. As a generalization of the critical exponent δ(Γ) of Γ we consider for any θ∈R≥0r​, ∥θ∥=1, the exponential growth rate δθ​(Γ) of the number of orbit points in X with prescribed „slope" θ. In analogy to Quint's result in [26] we show that the homogeneous extension ΨΓ​ to R≥0r​ of δθ​(Γ) as a function of θ 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 it spans. This shows in particular that there exists a unique slope θ∗ for which δθ∗​(Γ) is maximal and equal to the critical exponent of Γ.

Détail

DOI : 10.4171/ggd/370

Classification : 20-XX, 22-XX, 51-XX
Mots-clés : CAT(0)-spaces, products, cubical complexes, discrete groups, rank one isometries, limit set, limit cone, critical exponent