  We show that finitely generated groups which are Liouville and without
infinite finite-dimensional linear representations must have a global fixed
point whenever they act by isometry on a finite-dimensional complete
CAT(0)-space. This provides a partial answer to an old question in geometric
group theory and proves partly a conjecture formulated by Norin, Osajda, and
Przytycki. It applies in particular to Grigorchuk's groups of intermediate
growth and other branch groups as well as to simple groups with the Liouville
property such as those found by Matte Bon and by Nekrashevych. The method of
proof uses ultralimits, equivariant harmonic maps, subharmonic functions,
horofunctions and random walks.
