  This thesis is dedicated to the study of quasi-local boundary in quantum
gravity in the 3D space-time case. This research takes root in the holographic
principle, which conjectures that the geometry and the dynamic of a space-time
region can be entirely described by a theory living on the boundary of this
given region. The most studied case of this principle is the AdS/CFT
correspondence, where the quantum fluctuations of an asymptotically AdS space
are described by a conformal field theory living at spatial infinity, invariant
under the Virasoro group. The philosophy applied in this thesis differs from
the AdS/CFT case. I focus on the case of quasi-local holography, i.e. for a
bounded region of space-time with a boundary at a finite distance. The
objective is to clarify the bulk-boundary relation in quantum gravity described
by the Ponzano-Regge model, defining a model for 3D gravity via a discrete path
integral.
  I present the first perturbative and exact computations of the Ponzano-Regge
amplitude on a torus with a 2D boundary state. After the presentation of the
general framework for the 3D amplitude in terms of the 2D boundary state, I
consider the 2D torus case, with application in the study of the thermodynamics
of the BTZ black hole. First, the 2D boundary is described by a coherent spin
network state in the semi-classical regime. The stationary phase approximation
allows to recover in the asymptotic limit the usual amplitude for 3D quantum
gravity as the character of the symmetry of asymptotically flat gravity, the
BMS group. Then I introduce a new type of coherent boundary state, which allows
an exact evaluation of the amplitude for 3D quantum gravity. I obtain a complex
regularization of the BMS character. The possibility of this exact computation
suggests the existence of a (quasi)-integrable structure, linked to the
symmetries of 3D quantum gravity with 2D finite boundary.
