Constructing surfaces with first Steklov eigenvalue of arbitrarily large multiplicity

TL;DR

This paper constructs surfaces with arbitrarily large multiplicity for their first Steklov eigenvalue by using group actions and gluing techniques based on Cayley graphs, extending methods from Laplacian spectrum studies.

Contribution

It introduces a novel construction method for surfaces with high multiplicity Steklov eigenvalues using symmetry groups and geometric gluing techniques.

Findings

Surfaces with arbitrarily large multiplicity for first Steklov eigenvalue constructed.

Use of group actions and Cayley graph structures in surface construction.

Extension of Laplacian spectrum techniques to Steklov eigenvalue problem.

Abstract

We construct surfaces with arbitrarily large multiplicity for their first non-zero Steklov eigenvalue. The proof is based on a technique by M. Burger and B. Colbois originally used to prove a similar result for the Laplacian spectrum. We start by constructing surfaces $S_p$ with a specific subgroup of isometry $G_p:= \mathbb{Z}_p \rtimes \mathbb{Z}_p^*$ for each prime $p$. We do so by gluing surfaces with boundary following the structure of the Cayley graph of $G_p$. We then exploit the properties of $G_p$ and $S_p$ in order to show that an irreducible representation of high degree (depending on $p$) acts on the eigenspace of functions associated with $\sigma_1(S_p)$, leading to the desired result.