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Coboundary expansion inside Chevalley coset complex HDXs

Ryan O'Donnell, Noah G. Singer

Abstract

Recent major results in property testing~\cite{BLM24,DDL24} and PCPs~\cite{BMV24} were unlocked by moving to high-dimensional expanders (HDXs) constructed from $\widetilde{C}_d$-type buildings, rather than the long-known $\widetilde{A}_d$-type ones. At the same time, these building quotient HDXs are not as easy to understand as the more elementary (and more symmetric/explicit) \emph{coset complex} HDXs constructed by Kaufman--Oppenheim~\cite{KO18} (of $A_d$-type) and O'Donnell--Pratt~\cite{OP22} (of $B_d$-, $C_d$-, $D_d$-type). Motivated by these considerations, we study the $B_3$-type generalization of a recent work of Kaufman--Oppenheim~\cite{KO21}, which showed that the $A_3$-type coset complex HDXs have good $1$-coboundary expansion in their links, and thus yield $2$-dimensional topological expanders. The crux of Kaufman--Oppenheim's proof of $1$-coboundary expansion was: (1)~identifying a group-theoretic result by Biss and Dasgupta~\cite{BD01} on small presentations for the $A_3$-unipotent group over~$\mathbb{F}_q$; (2)~``lifting'' it to an analogous result for an $A_3$-unipotent group over polynomial extensions~$\mathbb{F}_q[x]$. For our $B_3$-type generalization, the analogue of~(1) appears to not hold. We manage to circumvent this with a significantly more involved strategy: (1)~getting a computer-assisted proof of vanishing $1$-cohomology of $B_3$-type unipotent groups over~$\mathbb{F}_5$; (2)~developing significant new ``lifting'' technology to deduce the required quantitative $1$-cohomology results in $B_3$-type unipotent groups over $\mathbb{F}_{5^k}[x]$.

Coboundary expansion inside Chevalley coset complex HDXs

Abstract

Recent major results in property testing~\cite{BLM24,DDL24} and PCPs~\cite{BMV24} were unlocked by moving to high-dimensional expanders (HDXs) constructed from -type buildings, rather than the long-known -type ones. At the same time, these building quotient HDXs are not as easy to understand as the more elementary (and more symmetric/explicit) \emph{coset complex} HDXs constructed by Kaufman--Oppenheim~\cite{KO18} (of -type) and O'Donnell--Pratt~\cite{OP22} (of -, -, -type). Motivated by these considerations, we study the -type generalization of a recent work of Kaufman--Oppenheim~\cite{KO21}, which showed that the -type coset complex HDXs have good -coboundary expansion in their links, and thus yield -dimensional topological expanders. The crux of Kaufman--Oppenheim's proof of -coboundary expansion was: (1)~identifying a group-theoretic result by Biss and Dasgupta~\cite{BD01} on small presentations for the -unipotent group over~; (2)~``lifting'' it to an analogous result for an -unipotent group over polynomial extensions~. For our -type generalization, the analogue of~(1) appears to not hold. We manage to circumvent this with a significantly more involved strategy: (1)~getting a computer-assisted proof of vanishing -cohomology of -type unipotent groups over~; (2)~developing significant new ``lifting'' technology to deduce the required quantitative -cohomology results in -type unipotent groups over .

Paper Structure

This paper contains 97 sections, 52 theorems, 474 equations, 8 figures, 7 tables.

Table of Contents

  1. Introduction
  2. Our main theorem
  3. Showing coboundary expansion
  4. Our new techniques, part I: Overcoming not being simply connected
  5. Our new techniques, part II: Lifting theorems
  6. Defining the group $U_{B_3}^{\mathrm{lg}}(\mathbb F_{q}[x]_{\leq 1})$
  7. The in-subgroup relations
  8. The lifted relations
  9. The game plan
  10. Preliminaries
  11. Simplicial complexes
  12. Expansion definitions
  13. Cocycle expansion
  14. Spectral expansion
  15. Topological expansion
  16. ...and 82 more sections

Key Result

theorem 1.4

(Main.) If $q$ is a sufficiently large power of $5$, the $2$-dimensional HDXs $\widehat{\mathfrak{B}}^{3}_{q}(m)$ are $(\epsilon_0,\mu_0)$-cosystolic expanders, where $\epsilon_0, \mu_0 > 0$ are universal constants.

Figures (8)

  • Figure 1: The root system $A_3$ and a specific "link" within it.
  • Figure 2: The root system $B_3$ and two specific "links" within it.
  • Figure 3: Establishing $\bm{\alpha+\beta+\gamma}$: A bipartite graph with left vertex-set $[2] \times [1]$ and right vertex-set $[1] \times [2]$, with an edge $(i,j) \sim (k,\ell)$ iff \ref{['rel:a3:interchange:alpha+beta+gamma']} (\ref{['rel:a3:interchange:alpha+beta+gamma']}) states that for all $t,u,v \in R$, $\left[{\bm{\alpha+\beta}tui}, {\bm{\gamma}vj}\right] = \left[{\bm{\alpha}tk}, {\bm{\beta+\gamma}2uv\ell}\right]$. (Hence, there are edges $(i+j,k) \sim (i,j+k)$ for every $i,j,k \in [1]$.) Additionally, grey blocks partition the vertices based on the sum of coordinates in $[3]$. In this case, the blocks also correspond to connected components in the graph.
  • Figure 4: Establishing $\bm{\beta+\psi+\omega}$: A bipartite graph with left vertex-set $[2] \times [1]$ and right vertex-set $[1] \times [2]$, with an edge $(i,j) \sim (k,\ell)$ iff \ref{['rel:b3-small:interchange:beta+psi+omega']} (\ref{['rel:b3-small:interchange:beta+psi+omega']}) states that for all $t,u,v \in R$, $\left[{\bm{\beta+\psi}tui}, {\bm{\omega}vj}\right] = \left[{\bm{\beta}tk}, {\bm{\psi+\omega}2uv\ell}\right]$. (Hence, there are edges $(i+j,k) \sim (i,j+k)$ for every $i,j,k \in [1]$.) Additionally, grey blocks partition the vertices based on the sum of coordinates in $[3]$. In this case, the blocks also correspond to connected components in the graph.
  • Figure 5: Establishing $\bm{\alpha+\beta+\psi}$: A bipartite graph with left vertex-set $[2] \times [1]$ and right vertex-set $[1] \times [2]$, with an edge $(i,j) \sim (k,\ell)$ only if for all $t,u,v \in R$, ${\bm{\psi}-v/2j} {\bm{\alpha+\beta}tui} {\bm{\psi}vj} {\bm{\alpha+\beta}-tui} {\bm{\psi}-v/2j} = {\bm{\beta+\psi}-uv/2\ell} {\bm{\alpha}tk} {\bm{\beta+\psi}uv\ell} {\bm{\alpha}-tk} {\bm{\beta+\psi}-uv/2\ell}$ from \ref{['rel:b3-large:raw:lift:alpha+beta+psi']} (\ref{['rel:b3-large:raw:lift:alpha+beta+psi']}). Additionally, grey blocks partition the vertices based on the sum of coordinates in $[3]$. In this case, the blocks also correspond to connected components in the graph.
  • ...and 3 more figures

Theorems & Definitions (246)

  • definition 1.1: The "$B_3$-type coset complex HDX"
  • remark 1.2
  • theorem 1.4
  • corollary 1.5
  • theorem 1.6
  • remark 1.7
  • definition 1.8
  • definition 2.1
  • definition 2.2
  • definition 2.4
  • ...and 236 more