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Spectral gap and origami expanders

Goulnara Arzhantseva, Dawid Kielak, Tim de Laat, Damian Sawicki

Abstract

We construct the first measure-preserving affine actions with spectral gap on surfaces of arbitrary genus $g > 1$. We achieve this by finding geometric representatives of multi-twists on origami surfaces. As a major application, we construct new expanders that are coarsely distinct from the classical expanders obtained via the Laplacian as Cayley graphs of finite quotients of a group. Our methods also show that the Margulis expander, and hence the Gabber--Galil expander, is coarsely distinct from the Selberg expander.

Spectral gap and origami expanders

Abstract

We construct the first measure-preserving affine actions with spectral gap on surfaces of arbitrary genus . We achieve this by finding geometric representatives of multi-twists on origami surfaces. As a major application, we construct new expanders that are coarsely distinct from the classical expanders obtained via the Laplacian as Cayley graphs of finite quotients of a group. Our methods also show that the Margulis expander, and hence the Gabber--Galil expander, is coarsely distinct from the Selberg expander.
Paper Structure (21 sections, 39 theorems, 56 equations, 4 figures)

This paper contains 21 sections, 39 theorems, 56 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Ergodicity and strong ergodicity
  4. Spectral gap and measure expansion
  5. Ahlfors-regular measures
  6. Lifting measure expansion to extensions
  7. Measure-expanding actions on origami surfaces
  8. An action of GL2(Z) on the torus
  9. A measure-expanding action of F_2 on T2
  10. Measure-expanding actions of F_2 on surfaces
  11. Staircases
  12. Large-scale geometry, warped cones, and expanders
  13. Uniform quasi-isometries and coarse embeddings
  14. Warped cones
  15. Selberg-type expanders and Margulis-type expanders
  16. ...and 6 more sections

Key Result

Theorem 1

Let $\Sigma$ be an arbitrary origami surface. Then the free group $\mathbb{F}_2$ admits a measure-preserving action by Lipschitz homeomorphisms on $\Sigma$ that is expanding in measure. Equivalently, this action $\mathbb{F}_2 \curvearrowright \Sigma$ has spectral gap.

Figures (4)

  • Figure 1: Constructing an origami surface of genus $2$
  • Figure 2: The staircase of genus 4
  • Figure 3: A local picture of the original Margulis expander $M_n$ (left) and the expander $M^\circ_n$ (right).
  • Figure 4: Staircase of genus 4 with labelled edges

Theorems & Definitions (107)

  • Definition 1.1: Spectral gap
  • Theorem 1
  • Corollary 2
  • Conjecture 3
  • Definition 1.2: Expander
  • Corollary 4
  • Theorem 5
  • Theorem 6
  • Theorem 7
  • Definition 2.1: Measure-preserving action
  • ...and 97 more