  We investigate classical spin systems in $d\geq 1$ dimensions whose transfer
operator commutes with the action of a nonamenable unitary representation of a
symmetry group, here ${\rm SO}(1,N)$; these systems may alternatively be
interpreted as systems of interacting quantum mechanical particles moving on
hyperbolic spaces. In sharp contrast to the analogous situation with a compact
symmetry group the following results are found and proven: (i) Spontaneous
symmetry breaking already takes place for finite spatial volume/finitely many
particles and even in dimensions $d=1,2$. The tuning of a coupling/temperature
parameter cannot prevent the symmetry breaking. (ii) The systems have
infinitely many non-invariant and non-normalizable generalized ground states.
(iii) the linear space spanned by these ground states carries a distinguished
unitary representation of ${\rm SO}(1,N)$, the limit of the spherical principal
series. (iv) The properties (i)--(iii) hold universally, irrespective of the
details of the interaction.
