Topology-changing amplitudes in 2 + 1 dimensional gravity

It has recently been observed that 2 + 1 dimensional gravity is a well-defined, finite, and soluble theory, and is particularly simple if the cosmological constant is zero. In this paper it is shown that in the latter case, it is possible to compute the topology-changing amplitudes rather explicitly. One finds that if the quantum theory is formulated on space-times that do not admit classical solutions, it is dominated by planckian distances. If formulated on space-times on which a classical solution is possible, the quantum theory escapes from the planckian domain into the classical regime. The cosmological constant, if zero in the classical lagrangian, is subject to neither finite nor infinite renormalization and remains zero in the quantum theory. We also briefly discuss the coupling to point masses, and some generalizations to include fermionic symmetries, including a super Chern-Simons action that is related to the Casson invariant.