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Swap cosystolic expansion

Yotam Dikstein, Irit Dinur

TL;DR

This work introduces swap cosystolic expansion, a novel high-dimensional expansion notion derived from the faces complex F^rX and the swap walk on r-faces. It develops a versatile toolkit—including non-abelian cones, GK decomposition, blow-ups, color-restriction techniques, and partitioned analysis of spherical buildings—to establish sub-exponential lower bounds on swap coboundary expansion for spherical buildings and on swap cosystolic expansion for LSV Ramanujan complexes, with general exp(-O(r)) bounds for broader coboundary expanders. The results connect to agreement testing in the low-acceptance regime, yielding unconditional agreement theorems for these complexes and reinforcing the link between cohomological expansion and robustness of constraint-satisfaction-style tests. The combination of color-based reductions, local-to-global principles, and inductive analysis on links provides a framework that advances sparse, derandomized high-dimensional expanders with practical implications for agreement tests and topological coding theory.

Abstract

We introduce and study swap cosystolic expansion, a new expansion property of simplicial complexes. We prove lower bounds for swap coboundary expansion of spherical buildings and use them to lower bound swap cosystolic expansion of the LSV Ramanujan complexes. Our motivation is the recent work (in a companion paper) showing that swap cosystolic expansion implies agreement theorems. Together the two works show that these complexes support agreement tests in the low acceptance regime. Swap cosystolic expansion is defined by considering, for a given complex $X$, its faces complex $F^r X$, whose vertices are $r$-faces of $X$ and where two vertices are connected if their disjoint union is also a face in $X$. The faces complex $F^r X$ is a derandomizetion of the product of $X$ with itself $r$ times. The graph underlying $F^rX$ is the swap walk of $X$, known to have excellent spectral expansion. The swap cosystolic expansion of $X$ is defined to be the cosystolic expansion of $F^r X$. Our main result is a $\exp(-O(\sqrt r))$ lower bound on the swap coboundary expansion of the spherical building and the swap cosystolic expansion of the LSV complexes. For more general coboundary expanders we show a weaker lower bound of $exp(-O(r))$.

Swap cosystolic expansion

TL;DR

This work introduces swap cosystolic expansion, a novel high-dimensional expansion notion derived from the faces complex F^rX and the swap walk on r-faces. It develops a versatile toolkit—including non-abelian cones, GK decomposition, blow-ups, color-restriction techniques, and partitioned analysis of spherical buildings—to establish sub-exponential lower bounds on swap coboundary expansion for spherical buildings and on swap cosystolic expansion for LSV Ramanujan complexes, with general exp(-O(r)) bounds for broader coboundary expanders. The results connect to agreement testing in the low-acceptance regime, yielding unconditional agreement theorems for these complexes and reinforcing the link between cohomological expansion and robustness of constraint-satisfaction-style tests. The combination of color-based reductions, local-to-global principles, and inductive analysis on links provides a framework that advances sparse, derandomized high-dimensional expanders with practical implications for agreement tests and topological coding theory.

Abstract

We introduce and study swap cosystolic expansion, a new expansion property of simplicial complexes. We prove lower bounds for swap coboundary expansion of spherical buildings and use them to lower bound swap cosystolic expansion of the LSV Ramanujan complexes. Our motivation is the recent work (in a companion paper) showing that swap cosystolic expansion implies agreement theorems. Together the two works show that these complexes support agreement tests in the low acceptance regime. Swap cosystolic expansion is defined by considering, for a given complex , its faces complex , whose vertices are -faces of and where two vertices are connected if their disjoint union is also a face in . The faces complex is a derandomizetion of the product of with itself times. The graph underlying is the swap walk of , known to have excellent spectral expansion. The swap cosystolic expansion of is defined to be the cosystolic expansion of . Our main result is a lower bound on the swap coboundary expansion of the spherical building and the swap cosystolic expansion of the LSV complexes. For more general coboundary expanders we show a weaker lower bound of .
Paper Structure (49 sections, 36 theorems, 162 equations, 3 figures)

This paper contains 49 sections, 36 theorems, 162 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Coboundaries and Unique Games Instances
  3. Coboundary and cosystolic expansion
  4. Our contribution
  5. Motivation: Agreement testing in the low acceptance regime
  6. Related work
  7. Open questions
  8. Preliminaries
  9. Probability distributions
  10. Expander graphs
  11. Graph homomorphisms and expansion
  12. Majority and expansion
  13. Local spectral expanders
  14. Coboundary and Cosystolic Expansion
  15. The faces complex
  16. ...and 34 more sections

Key Result

Lemma 1.2

Let $X$ be a graph and let $f\in C^1$. Then $f \in B^1$ if and only if the unique games instance defined by $f$ is strongly satisfiable.

Figures (3)

  • Figure 1: The cycle $P_u \circ (u,w) \circ P_w^{-1}$ and a filling of triangles.
  • Figure 2: A contraction for an edge $uw$
  • Figure 3: A triangle in $X$, in $\mathcal{A}$, and in the GK decomposition

Theorems & Definitions (147)

  • Definition 1.1
  • Lemma 1.2: See lem:cob-equiv-to-strong-sat for a slightly stronger statement
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Lemma 1.6
  • Corollary 1.7: Agreement for spherical buildings
  • Corollary 1.8: Agreement for LSV complexes
  • Definition 2.1
  • Claim 2.1.1
  • ...and 137 more