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On Renormalized Volume


Renormalized volume $\VR$ is a quantity that gives a notion of volume for hyperbolic manifolds having infinite volume under the classical definition. Its study for convex co-compact hyperbolic 3-manifolds can be found in \cite{KrasnovSchlenker}, while the geometrically finite case including rank 1-cusps was developed in \cite{MoroianuGuillarmouRochon}. In this dissertation we will answer what value is the infimum of the renormalized volume for a given geometrically finite hyperbolic manifold with incompressible boundary, and how a sequence converging to the infimum behaves. Partial results to this question were given in the acylindrical case by the author in \cite{Vargas16} and in the convex co-compact case by \cite{Vargas17} and \cite{BBB17} independently (\cite{BBB17} studied the gradient flow of the renormalized volume). In addition to finding the infimum value, we give the notion of additive geometric convergence in order to describe any sequence converging to the infimum. The study of local minima on the acylindrical case was done in parallel by \cite{Moroianu} and \cite{Vargas15}, which \cite{Vargas16} proves to be the infimum.

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