Asymptotic geometry in the product of Hadamard spaces

In this article we study asymptotic properties of certain discrete groups Γ acting by isometries on a product X = X1×X2 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 [Ben97] and [Qui02] hold in this context. In the first part of the paper we describe the structure of the geometric limit set of Γ and prove statements analogous to the results of Benoist in [Ben97]. The second part is concerned with the exponential growth rate δθ(Γ) of orbit points in X with a prescribed so-called ”slope” θ ∈ (0,π/2), which appropriately generalizes the critical exponent in higher rank. In analogy to Quint’s result in [Qui02] we show that the homogeneous extension ΨΓ to R≥0 of δθ(Γ) as a function of θ is upper semi-continuous and concave.