The category of crossed complexes gives an algebraic model of CW-complexes and cellular maps. Free crossed resolutions of groups contain information on a presentation of the group as well as higher homological information. We relate this to the problem of calculating non-abelian extensions. We show how the strong properties of this category allow for the computation of free crossed resolutions ...

Presenting 2-generator Artin groups A(m) and braid groups B3 and B4 as towers of HNN extensions of free groups, we obtain Gröbner–Shirshov bases, normal forms and rewriting systems for these groups. c © 2005 Elsevier Ltd. All rights reserved.

We introduce the notion of path extensions of tiling semigroups and investigate their properties. We show that the path extension of a tiling semigroup yields a strongly F *-inverse cover of the tiling semigroup and that it is isomorphic to an HNN * extension of its semilattice of idempo-tents.

In this paper, by using the Gröbner-Shirshov bases, we give characterizations of the Schreier extensions of groups when the group is presented by generators and relations. An algorithm to find the conditions of a group to be a Schreier extension is obtained. By introducing a special total order, we obtain the structure of the Schreier extension by an HNN group.

We study the property of tame combability for groups. We show that quasi-isometries preserve this property. We prove that an amalgamation, A ∗C B, where C is finitely generated, is tame combable iff both A and B are. An analogous result is obtained for HNN extensions. And we show that all one-relator groups are tame combable.