In this article we study asymptotic properties of certain discrete groups
acting by isometries
on a product
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
and prove statements analogous to the results of Benoist. The
second part is concerned with the exponential growth rate
of orbit points in
with a prescribed
“slope” ,
which appropriately generalizes the critical exponent in higher rank.
In analogy to Quint’s result we show that the homogeneous extension
to
of
as a
function of
is upper semicontinuous and concave.
Keywords
discrete group, $\mathrm{CAT}(0)$–spaces, limit set,
critical exponent, Kac–Moody groups
