  We explore the phase structure of nonlinear sigma models with target spaces
corresponding to compact quotients of hyperbolic space, focusing on the case of
a hyperbolic genus-2 Riemann surface. The continuum theory of these models can
be approximated by a lattice spin system which we simulate using Monte Carlo
methods. The target space possesses interesting geometric and topological
properties which are reflected in novel features of the sigma model. In
particular, we observe a topological phase transition at a critical
temperature, above which vortices proliferate, reminiscent of the
Kosterlitz-Thouless phase transition in the $O(2)$ model. Unlike in the $O(2)$
case, there are many different types of vortices, suggesting a possible analogy
to the Hagedorn treatment of statistical mechanics of a proliferating number of
hadron species. Below the critical temperature the spins cluster around six
special points in the target space known as Weierstrass points. The diversity
of compact hyperbolic manifolds suggests that our model is only the simplest
example of a broad class of statistical mechanical models whose main features
can be understood essentially in geometric terms.
