Upper bounds for Steklov eigenvalues of subgraphs of polynomial growth Cayley graphs

We study the Steklov problem on a subgraph with boundary $(\Omega,B)$ of a polynomial growth Cayley graph $\Gamma$. We prove that for each $k \in \mathbb{N}$, the $k^{\mbox{th}}$ eigenvalue tends to $0$ proportionally to $1/|B|^{\frac{1}{d-1}}$, where $d$ represents the growth rate of $\Gamma$. The method consists in associating a manifold $M$ to $\Gamma$ and a bounded domain $N \subset M$ to a subgraph $(\Omega, B)$ of $\Gamma$. We find upper bounds for the Steklov spectrum of $N$ and transfer these bounds to $(\Omega, B)$ by discretizing $N$ and using comparison Theorems.