Leighton's Theorem and Regular Cube Complexes

TL;DR

This paper generalizes Leighton's theorem from finite graphs to a broad class of non-positively curved special cube complexes, including hyperbolic and non-hyperbolic CAT(0) complexes, expanding the theorem's applicability.

Contribution

It extends Leighton's graph covering theorem to non-positively curved special cube complexes, broadening its scope to higher-dimensional structures.

Findings

Generalization of Leighton's theorem to cube complexes

Includes both hyperbolic and non-hyperbolic CAT(0) complexes

Establishes conditions for common finite covers in complex families

Abstract

Leighton's graph covering theorem states that two finite graphs with common universal cover have a common finite cover. We generalize this to a large family of non-positively curved special cube complexes that form a natural generalization of regular graphs. This family includes both hyperbolic and non-hyperbolic CAT(0) cube complexes.