Asymptotic geometry in the product of Hadamard spaces

In this article we study asymptotic properties of certain discrete groups Γ act-ing 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, be-long 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 R2≥0 of δθ(Γ) as a function of θ is upper semi-continuous and concave.