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Coboundary and cosystolic expansion without dependence on dimension or degree

Yotam Dikstein, Irit Dinur

TL;DR

A new color-restriction technique is developed which enables proving dimension-free expansion by restricting a multi-partite complex to small random subsets of its color classes, and a new spectral proof for Evra and Kaufman's local-to-global theorem is given, deriving better bounds and getting rid of the dependence on the degree.

Abstract

We give new bounds on the cosystolic expansion constants of several families of high dimensional expanders, and the known coboundary expansion constants of order complexes of homogeneous geometric lattices, including the spherical building of $SL_n(F_q)$. The improvement applies to the high dimensional expanders constructed by Lubotzky, Samuels and Vishne, and by Kaufman and Oppenheim. Our new expansion constants do not depend on the degree of the complex nor on its dimension, nor on the group of coefficients. This implies improved bounds on Gromov's topological overlap constant, and on Dinur and Meshulam's cover stability, which may have applications for agreement testing. In comparison, existing bounds decay exponentially with the ambient dimension (for spherical buildings) and in addition decay linearly with the degree (for all known bounded-degree high dimensional expanders). Our results are based on several new techniques: * We develop a new "color-restriction" technique which enables proving dimension-free expansion by restricting a multi-partite complex to small random subsets of its color classes. * We give a new "spectral" proof for Evra and Kaufman's local-to-global theorem, deriving better bounds and getting rid of the dependence on the degree. This theorem bounds the cosystolic expansion of a complex using coboundary expansion and spectral expansion of the links. * We derive absolute bounds on the coboundary expansion of the spherical building (and any order complex of a homogeneous geometric lattice) by constructing a novel family of very short cones.

Coboundary and cosystolic expansion without dependence on dimension or degree

TL;DR

A new color-restriction technique is developed which enables proving dimension-free expansion by restricting a multi-partite complex to small random subsets of its color classes, and a new spectral proof for Evra and Kaufman's local-to-global theorem is given, deriving better bounds and getting rid of the dependence on the degree.

Abstract

We give new bounds on the cosystolic expansion constants of several families of high dimensional expanders, and the known coboundary expansion constants of order complexes of homogeneous geometric lattices, including the spherical building of . The improvement applies to the high dimensional expanders constructed by Lubotzky, Samuels and Vishne, and by Kaufman and Oppenheim. Our new expansion constants do not depend on the degree of the complex nor on its dimension, nor on the group of coefficients. This implies improved bounds on Gromov's topological overlap constant, and on Dinur and Meshulam's cover stability, which may have applications for agreement testing. In comparison, existing bounds decay exponentially with the ambient dimension (for spherical buildings) and in addition decay linearly with the degree (for all known bounded-degree high dimensional expanders). Our results are based on several new techniques: * We develop a new "color-restriction" technique which enables proving dimension-free expansion by restricting a multi-partite complex to small random subsets of its color classes. * We give a new "spectral" proof for Evra and Kaufman's local-to-global theorem, deriving better bounds and getting rid of the dependence on the degree. This theorem bounds the cosystolic expansion of a complex using coboundary expansion and spectral expansion of the links. * We derive absolute bounds on the coboundary expansion of the spherical building (and any order complex of a homogeneous geometric lattice) by constructing a novel family of very short cones.
Paper Structure (65 sections, 30 theorems, 138 equations, 2 figures, 3 algorithms)

This paper contains 65 sections, 30 theorems, 138 equations, 2 figures, 3 algorithms.

Table of Contents

  1. Introduction
  2. Applications of cosystolic expansion
  3. Topological overlap
  4. Cover stability
  5. Agreement testing
  6. Related work
  7. Open questions
  8. Overview of the proof of thm:main-LSV
  9. The local to global argument, thm:cosystolic-expansion-clear
  10. The "color restriction" technique, thm:color-restriction
  11. New bounds on color-restrictions of the spherical building via cones, thm:cones-intro
  12. Organization of this paper
  13. Acknowledgment
  14. Preliminaries and notation
  15. Simplicial complexes
  16. ...and 50 more sections

Key Result

Theorem 1.1

For every integer $d > 1$ and every small enough $\lambda > 0$ let $X$ be a $d$-dimensional LSV complex whose links are $\lambda$-one-sided expanders. For every group The theorem holds for every group $\Gamma$ for which cohomology is defined, namely, abelian groups for $k>1$ and any group for $k=1$.

Figures (2)

  • Figure 1: Tiling a cycle
  • Figure 2: Constructing $\psi(s)$

Theorems & Definitions (81)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Claim 2.1
  • Definition 2.2
  • Claim 2.3
  • proof : Proof of claim:edge-expander-many-sets
  • Definition 2.4: Spectral expander
  • Definition 2.5: high dimensional local spectral expander
  • ...and 71 more