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

Abstract

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.

The Steklov Problem on Triangle-Tiling Graphs in the Hyperbolic Plane

“…As said before, we want to work on graphs that have exponential growth rates; therefore, we will only consider the third case in this paper. Since one may ask if our result is still true for the two other cases, we remark that in the Euclidean case, the triangle group has polynomial growth rate and then has already been studied in [16]. Regarding the spherical case, the triangle group is finite, and hence, one can theoretically compute all different possible situations.…”

“…As a corollary, we have Corollary 7 (Corollary 6 in [16]) Let be a polynomial growth Cayley graph of order d ≥ 2 and ( l ) ∞ l=1 be a sequence of subgraphs of such that…”

“…This theorem is way more general about the class of host graph but provides us control over the first non-trivial eigenvalue only, see [13] for details. We gave an extension to this result in a precedent article: Theorem 6 (Theorem 5 in [16]) Let = Cay(G, S) be a polynomial growth Cayley graph of order d ≥ 2. Let be a subgraph of .…”

The First Steklov Eigenvalue on Infinite Subgraphs of Integer Lattices

“…In [15], the authors prove a uniform spectral comparison inequality between the Steklov eigenvalues of a manifold and those of its discretization. This idea used by Tschanz in [36] to transfer upper bounds of the Steklov spectrum of a bounded domain of M to subgraph of finite polynomial growth Cayley graphs. This strategy is known as coarse discretization, which is used in the literature to the spectral geometry of the Laplacian on closed Riemannian manifolds.…”

Minimal Steklov eigenvalues on combinatorial graphs

“…Although the notions are new, there have been a number of studies on exploring the properties of Steklov eigenvalues in the discrete setting. For example, in [13,16,17,12,14,20,26], the authors studied isoperimetric control of Steklov eigenvalues. In [15,27], the authors studied monotonicity of Steklov eigenvalues.…”