Imitator homomorphisms for special cube complexes

TL;DR

This paper introduces imitator homomorphisms as a new perspective on the canonical completion in special cube complexes, extending key theorems to non-hyperbolic cases and proving a convex omnipotence result.

Contribution

It offers a novel interpretation of the canonical completion using imitator homomorphisms, generalizing Haglund--Wise results to non-hyperbolic settings.

Findings

New interpretation of canonical completion via imitator homomorphisms

Generalization of existing theorems to non-hyperbolic special cube complexes

Proof of a convex version of omnipotence for virtually special cubulated groups

Abstract

Central to the theory of special cube complexes is Haglund and Wise's construction of the canonical completion and retraction, which enables one to build finite covers of special cube complexes in a highly controlled manner. In this paper we give a new interpretation of this construction using what we call imitator homomorphisms. This provides fresh insight into the construction and enables us to prove various new results about finite covers of special cube complexes -- most of which generalise existing theorems of Haglund--Wise to the non-hyperbolic setting. In particular, we prove a convex version of omnipotence for virtually special cubulated groups.