Asymptotic Geometry in the product of Hadamard spaces with rank one isometries

TL;DR

This paper investigates the asymptotic geometry of discrete groups acting on products of Hadamard spaces, extending classical higher rank properties to Kac-Moody groups and analyzing growth rates with respect to slopes.

Contribution

It describes the structure of the geometric limit set and establishes properties of the exponential growth rate function for groups acting on product Hadamard spaces.

Findings

Description of the geometric limit set structure.

Proven upper semi-continuity and concavity of the growth rate function.

Analogues of classical higher rank properties for Kac-Moody groups.

Abstract

In this article we study asymptotic properties of certain discrete groups $\Gamma$ acting by isometries on a product $\XX=\XX_1\times \XX_2$ of locally compact Hadamard spaces. The motivation comes from the fact that Kac-Moody groups over finite fields, which can be seen as generalizations of arithmetic groups over function fields, belong to this class of groups. Hence one may ask whether classical properties of discrete subgroups of higher rank Lie groups as in [MR1437472] and [MR1933790] hold in this context. In the first part of the paper we describe the structure of the geometric limit set of $\Gamma$ and prove statements analogous to the results of Benoist in [MR1437472]. The second part is concerned with the exponential growth rate $\delta_\theta(\Gamma)$ of orbit points in $\XX$ with a prescribed so-called "slope" $\theta\in (0,\pi/2)$, which appropriately generalizes the critical exponent in higher rank. In analogy to Quint's result in [MR1933790] we show that the homogeneous extension $\Psi_\Gamma$ to $\RR_{\ge 0}^2$ of $\delta_\theta(\Gamma)$ as a function of $\theta$ is upper semi-continuous and concave.