Imitator homomorphisms for special cube complexes
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.
Cite
@article{arxiv.2107.10925,
title = {Imitator homomorphisms for special cube complexes},
author = {Sam Shepherd},
journal= {arXiv preprint arXiv:2107.10925},
year = {2022}
}
Comments
45 pages, 6 figures; v2: Lemma 7.3 has become Theorem 1.6, a mistake corrected in the proof of Theorem 1.7, the statement of Theorem 1.8 has been clarified, plus other smaller changes; to appear in Transactions of the AMS