# Coboundary and cosystolic expansion from strong symmetry

## TL;DR

This paper investigates high-dimensional expansion properties of simplicial complexes with strong symmetry, introducing new methods to establish coboundary and cosystolic expansion beyond classical spectral analysis.

## Contribution

It develops a novel machinery leveraging strong symmetry to prove coboundary and cosystolic expansion in high-dimensional complexes, expanding understanding beyond known building-based examples.

## Key findings

- Established coboundary and cosystolic expansion for strongly symmetric complexes.
- Provided new tools for analyzing high-dimensional expansion properties.
- Extended the class of known high-dimensional expanders beyond building-based constructions.

## Abstract

Coboundary and cosystolic expansion are notions of expansion that generalize the Cheeger constant or edge expansion of a graph to higher dimensions. The classical Cheeger inequality implies that for graphs edge expansion is equivalent to spectral expansion. In higher dimensions this is not the case: a simplicial complex can be spectrally expanding but not have high dimensional edge-expansion. The phenomenon of high dimensional edge expansion in higher dimensions is much more involved than spectral expansion, and is far from being understood. In particular, prior to this work, the only known bounded degree cosystolic expanders known were derived from the theory of buildings that is far from being elementary. In this work we study high dimensional complexes which are {\em strongly symmetric}. Namely, there is a group that acts transitively on top dimensional cells of the simplicial complex [e.g., for graphs it corresponds to a group that acts transitively on the edges]. Using the strong symmetry, we develop a new machinery to prove coboundary and cosystolic expansion.