  We discuss a formalism for solving (2+1) AdS gravity on Riemann surfaces. In
the torus case the equations of motion are solved by two functions f and g,
solutions of two independent O(2,1) sigma models, which are distinct because
their first integrals contain a different time dependent phase factor. We then
show that with the gauge choice $k = \sqrt{\Lambda}/ tg (2 \sqrt{\Lambda}t)$
the same couple of first integrals indeed solves exactly the Einstein equations
for every Riemann surface. The $X^A=X^A(x^mu)$ polydromic mapping which extends
the standard immersion of a constant curvature three-dimensional surface in a
flat four-dimensional space to the case of external point sources or topology,
is calculable with a simple algebraic formula in terms only of the two sigma
model solutions f and g. A trivial time translation of this formalism allows us
to introduce a new method which is suitable to study the scattering of black
holes in (2+1) AdS gravity.
