Triangulating 3D gravity

I will discuss recent work on the path integral of pure 3D gravity on a finite region of spacetime with boundary conditions that fix dihedral angles or geodesic lengths. In the dual CFT, this amplitude calculates statistics of black hole matrix elements. The fixed-length path integral is related to Virasoro TQFT, while the fixed-angle path integral is the partition function of Conformal Turaev-Viro theory, a topological theory based on triangulations. This gives a natural way to understand the exact gravity path integral as well as the wormhole geometries that appear in the semiclassical limit.