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This thesis is dedicated to the study of quasi-local boundary in quantumgravity in the 3D space-time case. This research takes root in the holographicprinciple, which conjectures that the geometry and the dynamic of a space-timeregion can be entirely described by a theory living on the boundary of thisgiven region. The most studied case of this principle is the AdS/CFTcorrespondence, where the quantum fluctuations of an asymptotically AdS spaceare described by a conformal field theory living at spatial infinity, invariantunder the Virasoro group. The philosophy applied in this thesis differs fromthe AdS/CFT case. I focus on the case of quasi-local holography, i.e. for abounded region of space-time with a boundary at a finite distance. Theobjective is to clarify the bulk-boundary relation in quantum gravity describedby the Ponzano-Regge model, defining a model for 3D gravity via a discrete pathintegral.  I present the first perturbative and exact computations of the Ponzano-Reggeamplitude on a torus with a 2D boundary state. After the presentation of thegeneral framework for the 3D amplitude in terms of the 2D boundary state, Iconsider the 2D torus case, with application in the study of the thermodynamicsof the BTZ black hole. First, the 2D boundary is described by a coherent spinnetwork state in the semi-classical regime. The stationary phase approximationallows to recover in the asymptotic limit the usual amplitude for 3D quantumgravity as the character of the symmetry of asymptotically flat gravity, theBMS group. Then I introduce a new type of coherent boundary state, which allowsan exact evaluation of the amplitude for 3D quantum gravity. I obtain a complexregularization of the BMS character. The possibility of this exact computationsuggests the existence of a (quasi)-integrable structure, linked to thesymmetries of 3D quantum gravity with 2D finite boundary.

Quasi-Local 3D Quantum Gravity : Exact Amplitude and Holography