  We generalize the fixed-point property for discrete groups acting on convex
cones given by Monod in \cite{monod} to topological groups. At first, we focus
on describing this fixed-point property from a functional point of view, and
then we look at the class of groups that have it. Finally, we go through some
applications of this fixed-point property. To accomplish these tasks, we
introduce a new class of normed Riesz spaces that depend on group
representation.
