We introduce the notion of coupled link cube complex (CLCC) as a means constructing interesting cocompactly cubulated groups. CLCCs are defined locally, making them useful tool when precise control over links is required. In this paper we study some general properties CLCCs, such their (co)homological dimension and criteria for hyperbolicity. Some examples fundamental groups RAAGs, RACGs, surfa...

We give new examples of hyperbolic and relatively hyperbolic groups of cohomological dimension d for all d ≥ 4 (see Theorem 2.13). These examples result from applying CAT(0)/CAT(−1) filling constructions (based on singular doubly warped products) to finite volume hyperbolic manifolds with toral cusps. The groups obtained have a number of interesting properties, which are established by analyzin...

We prove that every finite-volume hyperbolic 3-manifold M with p ≥ 1 cusps admits a canonical, complete, piecewise Euclidean CAT(0) metric, with a canonical projection to a CAT(0) spine K∗ M . Moreover: (a) The universal cover of M endowed with the CAT(0) metric is a union of Euclidean half-spaces, glued together by identifying Euclidean polygons in their bounding planes by pairwise isometry; (...