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We study non-perturbative quantization of 3d gravity with positivecosmological constant (de Sitter space being the prototype vacuum solution,whose Euclideanization of course gives the three sphere) on the backgroundtopology of lens space, which is a three spheres modulo a discrete group.Instead of the strategy followed by a recent work \cite{Castro:2011xb}, whichcompares results in the second and first order formulations of gravity, weconcentrate on the later solely. We note, as a striking feature, that thequantization, that relies heavily on the axiomatics of topological quantumfield theory (TQFT) can only be consistently carried by augmenting theconventional theory by an additional topological term coupled through adimensionless parameter. More importantly the introduction of this additionalparameter renders the theory finite.

Consistent 3D Quantum Gravity on Lens Spaces