A new construction of CAT(0) cube complexes

TL;DR

This paper introduces coupled link cube complexes (CLCCs), a new local construction method for creating cocompactly cubulated groups, with applications to hyperbolic 3- and 4-manifolds.

Contribution

It presents the concept of CLCCs, analyzes their properties, and demonstrates their utility in constructing hyperbolic cubulated manifolds and groups.

Findings

CLCCs are useful for controlling links in cube complexes.

Criteria for hyperbolicity of CLCCs are established.

Examples include RAAGs, RACGs, surface groups, and manifold groups.

Abstract

We introduce the notion of coupled link cube complex (CLCC) as a means of constructing interesting cocompactly cubulated groups. CLCCs are defined locally, making them a useful tool when precise control over the links is required. In this paper we study some general properties of CLCCs, such as their (co)homological dimension and criteria for hyperbolicity. Some examples of fundamental groups of CLCCs are RAAGs, RACGs, surface groups and some manifold groups. As immediate applications of our criteria we produce a number of cubulated 3 and 4 manifolds with hyperbolic fundamental group.