  We use the analytic continuation procedure proposed in our earlier works to
study the thermodynamics of black holes in 2+1 dimensions. A general black hole
in 2+1 dimensions has g handles hidden behind h horizons. The result of the
analytic continuation is a hyperbolic 3-manifold having the topology of a
handlebody. The boundary of this handlebody is a compact Riemann surface of
genus G=2g+h-1. Conformal moduli of this surface encode in a simple way the
physical characteristics of the black hole. The moduli space of black holes of
a given type (g,h) is then the Schottky space at genus G. The (logarithm of
the) thermodynamic partition function of the hole is the Kaehler potential for
the Weil-Peterson metric on the Schottky space. Bekenstein bound on the black
hole entropy leads us to conjecture a new strong bound on this Kaehler
potential.
