  We study the one-loop partition function of 3D gravity without cosmological
constant on the solid torus with arbitrary metric fluctuations on the boundary.
To this end we employ the discrete approach of (quantum) Regge calculus. In
contrast with similar calculations performed directly in the continuum, we work
with a boundary at finite distance from the torus axis. We show that after
taking the continuum limit on the boundary - but still keeping finite distance
from the torus axis - the one-loop correction is the same as the one recently
found in the continuum in Barnich et al. for an asymptotically flat boundary.
The discrete approach taken here allows to identify the boundary degrees of
freedom which are responsible for the non-trivial structure of the one-loop
correction. We therefore calculate also the Hamilton-Jacobi function to
quadratic order in the boundary fluctuations both in the discrete set-up and
directly in the continuum theory. We identify a dual boundary field theory with
a Liouville type coupling to the boundary metric. The discrete set-up allows
again to identify the dual field with degrees of freedom associated to radial
bulk edges attached to the boundary. Integrating out this dual field reproduces
the (boundary diffeomorphism invariant part of the) quadratic order of the
Hamilton-Jacobi functional. The considerations here show that bulk boundary
dualities might also emerge at finite boundaries and moreover that discrete
approaches are helpful in identifying such dualities.
