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Isospectrality for infinite-type hyperbolic surfaces with discrete length spectrum

Federica Fanoni, David Fisac

Abstract

We prove that every family of isospectral surfaces with discrete length spectrum arising from Sunada's method is finite. Furthermore, by introducing the topological notion of surfaces with self-duplicating ends, we show that every finite group can be realized as the full isometry group of a hyperbolic structure with discrete spectrum on such a surface, if the genus is infinite. Under the same topological assumptions, we also demonstrate that the above-mentioned isospectral families can have unbounded cardinality within a fixed moduli space.

Isospectrality for infinite-type hyperbolic surfaces with discrete length spectrum

Abstract

We prove that every family of isospectral surfaces with discrete length spectrum arising from Sunada's method is finite. Furthermore, by introducing the topological notion of surfaces with self-duplicating ends, we show that every finite group can be realized as the full isometry group of a hyperbolic structure with discrete spectrum on such a surface, if the genus is infinite. Under the same topological assumptions, we also demonstrate that the above-mentioned isospectral families can have unbounded cardinality within a fixed moduli space.
Paper Structure (7 sections, 12 theorems, 17 equations, 3 figures)

This paper contains 7 sections, 12 theorems, 17 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Group theory
  4. Finiteness of isospectral families given by Sunada's construction
  5. Realizing finite groups as isometry groups
  6. Arbitrarily large isospectral families
  7. Limits of isospectral families

Key Result

Theorem 1.1

Let $X$ be a complete hyperbolic surface, and $G$ a group of isometries with finitely many fixed points. For $H_1,H_2$ two almost conjugate subgroups of $G$ acting without fixed points, the quotient surfaces $X/H_1$ and $X/H_2$ are isospectral.

Figures (3)

  • Figure 1: Blooming Cantor tree surface.
  • Figure 2: Construction of $X$ for $G=\mathbb{Z}/3\mathbb{Z}$ and $S$ the blooming Cantor tree surface.
  • Figure 3: Construction of $\mathcal{P}$.

Theorems & Definitions (22)

  • Theorem 1.1: Sun_AlmostConjugate
  • Theorem A
  • Theorem B
  • Theorem C
  • Theorem 1.2: APV_Isometry
  • Theorem 2.1: Collar Lemma
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • ...and 12 more