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

We study the Steklov problem on a subgraph with boundary $$(\Omega ,B)$$ ( Ω , B ) of a polynomial growth Cayley graph $$\Gamma$$ Γ . For $$(\Omega _l, B_l)_{l=1}^\infty$$ ( Ω l , B l ) l = 1 ∞ a sequence of subgraphs of $$\Gamma$$ Γ such that $$|\Omega _l| \longrightarrow \infty$$ | Ω l | ⟶ ∞ , we prove that for each $$k \in {\mathbb {N}}$$ k ∈ N , the kth eigenvalue tends to 0 proportionally to $$1/|B|^{\frac{1}{d-1}}$$ 1 / | B | 1 d - 1 , where d represents the growth rate of $$\Gamma$$ Γ . The method consists in associating a manifold M to $$\Gamma$$ Γ and a bounded domain $$N \subset M$$ N ⊂ M to a subgraph $$(\Omega , B)$$ ( Ω , B ) of $$\Gamma$$ Γ . We find upper bounds for the Steklov spectrum of N and transfer these bounds to $$(\Omega , B)$$ ( Ω , B ) by discretizing N and using comparison theorems.