Asymptotic geometry in products of Hadamard spaces with rank one isometries

Abstract

In this article we study asymptotic properties of certain discrete groups $\Gamma$ acting by isometries on a product $X=X_1\times X_2$ of locally compact Hadamard spaces which admit a geodesic without flat half-plane. 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 the considered class of groups. Hence one may ask whether classical properties of discrete subgroups of higher rank Lie groups as in Benoist [Geom. Funct. Anal. 7 (1997) 1-47] and Quint [Comment. Math. Helv. 77 (2002) 563-608] 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. The second part is concerned with the exponential growth rate ${\delta }_{\theta }\left(\Gamma \right)$ of orbit points in $X$ with a prescribed “slope” $\theta \in \left(0,\pi ∕2\right)$, which appropriately generalizes the critical exponent in higher rank. In analogy to Quint’s result we show that the homogeneous extension ${\Psi }_{\Gamma }$ to ${ℝ}_{\ge 0}^2$ of ${\delta }_{\theta }\left(\Gamma \right)$ as a function of $\theta$ is upper semicontinuous and concave.