Folding-like techniques for CAT(0) cube complexes

TL;DR

This paper extends Stallings-like folding techniques from free groups and right-angled Coxeter groups to fundamental groups of non-positively curved cube complexes, enabling new algorithmic approaches.

Contribution

It introduces novel folding-like methods tailored for CAT(0) cube complexes, broadening the applicability of Stallings's techniques.

Findings

Developed folding techniques for CAT(0) cube complexes

Provided algorithmic tools for subgroup analysis in non-positively curved spaces

Extended existing methods to a broader class of groups

Abstract

In a seminal paper, Stallings introduced folding of morphisms of graphs. One consequence of folding is the representation of finitely-generated subgroups of a finite-rank free group as immersions of finite graphs. Stallings's methods allow one to construct this representation algorithmically, giving effective, algorithmic answers and proofs to classical questions about subgroups of free groups. Recently Dani--Levcovitz used Stallings-like methods to study subgroups of right-angled Coxeter groups, which act geometrically on CAT(0) cube complexes. In this paper we extend their techniques to fundamental groups of non-positively curved cube complexes.