The Steklov spectrum and coarse discretizations of manifolds with boundary

Given $\kappa, r_0>0$ and $n\in\N$, we consider the class
$\mathcal{M}=\mathcal{M}(\kappa,r_0,n)$ of compact $n$-dimensional
Riemannian manifolds with cylindrical boundary, Ricci curvature
bounded below by $-(n-1)\kappa$ and injectivity radius bounded below
by $r_0$ away from the boundary. For a manifold $M\in\mathcal{M}$ we introduce a notion of
discretization, leading to a graph with boundary which is roughly
isometric to $M$, with constants depending only on $\kappa,r_0,n$. In
this context, we prove a uniform spectral comparison inequality
between the Steklov eigenvalues of a manifold $M\in\mathcal{M}$ 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 $\sigma_2-\sigma_1$ 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.