Three-dimensional gravity from the Turaev-Viro invariant

We study the q-deformed su(2) spin network as a three-dimensional quantum gravity model. We show that in the semiclassical continuum limit the Turaev-Viro invariant obtained recently defines a naturally regularized path integral a` la Ponzano and Regge, in which a contribution from the cosmological term is effectively included. The regularization-dependent cosmological constant is found to be 4${\mathrm{\pi }}^2$/${\mathit{k}}^2$+O(${\mathit{k}}^{\mathrm{-}4}$), where ${\mathit{q}}^{2\mathit{k}}$=1. We also discuss the relation to the Euclidean Chern-Simons-Witten gravity in three dimensions.