  For finitely supported random walks on finitely generated groups $G$ we prove
that the identity map on $G$ extends to a continuous equivariant surjection
from the Martin boundary to the Floyd boundary, with preimages of conical
points being singletons. This yields new results for relatively hyperbolic
groups. Our key estimate relates the Green and Floyd metrics, generalizing
results of Ancona for random walks on hyperbolic groups and of Karlsson for
quasigeodesics.
  We then apply these techniques to obtain some results concerning the harmonic
measure on the limit sets of geometrically finite isometry groups of Gromov
hyperbolic spaces. .
