In this paper we are interested in finite connected 2-dimensional CTF-complexes, each with a single 2-celL We show any two such complexes have the same homotopy type if their fundamental groups are isomorphic. In fact, there is a homotopy equivalence inducing any isomorphism of the fundamental groups. We also study the homotopy factorizations of such spaces into finite sums. In this paper we ar...

A criterion for the HN N-extension of a finite p-group to be residually a finite p-group is obtained and based on this criterion the sufficient condition for residuality a finite p-group of HN N-extension with arbitrary base group is proved. Then these results are applied to give for groups from two classes of one-relator groups the necessary and sufficient condition to be residually a finite p...

In the article, a result relating to maps (= finite planar connected and simply connected 2-complexes) that satisfy a C(p)&T (q) condition (where (p, q) is one of (3, 6), (4, 4), (6, 3) which correspond to regular tessellations of the plane by triangles, squares, hexagons, respectively) is proven. On the base of this result a criterion for the Gromov hyperbolicity of finitely presented small ca...

Duncan, A.J. and J. Howie, Spelling theorems and Cohen-Lyndon theorems for one-relator products, Journal of Pure and Applied Algebra 92 (1994) 123-136. Spelling theorems, a Cohen-Lyndon theorem and a Magnus theorem are proved for onerelator products of arbitrary groups, in cases where the relator is a sufficiently high power.

We show that the free-by-cyclic groups of the form F2 o Z act properly cocompactly on CAT(0) square complexes. We also show using generalized Baumslag–Solitar groups that all known groups defined by a 2-generator 1-relator presentation are either SQ-universal or are cyclic or isomorphic to a soluble Baumslag–Solitar group.

Pride groups, or “groups given by presentations in which each defining relator involves at most two types of generators”, include Coxeter groups, Artin groups, triangles of groups, and Vinberg’s groups defined by periodic paired relations. We show that every non-spherical Pride group that is not a triangle of groups satisfies the Tits alternative.

We give a lower and an upper bound for the conformal dimension of the boundaries of certain small cancellation groups. We apply these bounds to the few relator and density models for random groups. This gives generic bounds of the following form, where l is the relator length, going to infinity. (a) 1 + 1/C < Cdim(∂∞G) < Cl/ log(l), for the few relator model, and (b) 1 + l/(C log(l)) < Cdim(∂∞G...

If K = Goφ Z where φ is a tame automorphism of the 1-relator group G, then the combinatorial area of loops in a Cayley graph of G is undistorted in a Cayley graph of K. Examples of distortion of area in fibres of fibrations over the circle are given and a notion of exponent of area distortion is introduced and studied. The inclusion of a finitely generated abelian subgroup in the fundamental gr...