  Benjamini and Schramm introduced the notion of distributional limit of a
sequence of graphs with uniformly bounded valence and studied such limits in
the case that the involved graphs are planar. We investigate distributional
limits of sequences of Riemannian manifolds with bounded curvature which
satisfy certain condition of quasi-conformal nature. We then apply our results
to somewhat improve Benjamini's and Schramm's original result on the recurrence
of the simple random walk on limits of planar graphs. For instance, as an
application give a proof of the fact that for graphs in an expander family, the
genus of each graph is bounded from below by a linear function of the number of
vertices.
