  We consider the class of compact n-dimensional Riemannian manifolds with
cylindrical boundary, Ricci curvature bounded below by a given constant and
injectivity radius bounded below by a positive constant, away from the
boundary. For a manifold M of this class, we introduce a notion of
discretization, leading to a graph with boundary which is roughly isometric to
M, with constants depending only on the dimension and bounds on curvature and
injectivity radius. In this context, we prove a uniform spectral comparison
inequality between the Steklov eigenvalues of the manifold M and those of its
discretization. Some applications to the construction of sequences of surfaces
with boundary of fixed length and with arbitrarily large Steklov spectral gap
are given. In particular, we obtain such a sequence for surfaces with connected
boundary. The applications are based on the construction of graph-like surfaces
which are obtained from sequences of graphs with good expansion properties.
