Convergence of the barycentre of measures from Fuchsian action groups

Abstract

We prove the pointwise ergodic convergence of the sequence of barycentres of empirical measures which are defined from the action of Fuchsian groups and by maps valuated in CAT(0)−spaces. A result of this nature was established by Austin from actions of amenable groups and defining the empirical measures from Følner sequences. Here we de- fine different sequences of barycentres, in particular we do not consider a topological structure on the group and Følner sequences.