Discrete Morse theory and graph braid groups

If Gamma is any finite graph, then the unlabelled configuration space of n points on Gamma, denoted UC^n(Gamma), is the space of n-element subsets of Gamma. The braid group of Gamma on n strands is the fundamental group of UC^n(Gamma). We apply a discrete version of Morse theory to these UC^n(Gamma), for any n and any Gamma, and provide a clear description of the critical cells in every case. As a result, we can calculate a presentation for the braid group of any tree, for any number of strands. We also give a simple proof of a theorem due to Ghrist: the space UC^n(Gamma) strong deformation retracts onto a CW complex of dimension at most k, where k is the number of vertices in Gamma of degree at least 3 (and k is thus independent of n).Comment: Published by Algebraic and Geometric Topology at http://www.maths.warwick.ac.uk/agt/AGTVol5/agt-5-44.abs.htm

On groups that have normal forms computable in logspace

We consider the class of finitely generated groups which have a normal form computable in logspace. We prove that the class of such groups is closed under finite extensions, finite index subgroups, direct products, wreath products, and also certain free products, and includes the solvable Baumslag-Solitar groups, as well as non-residually finite (and hence non-linear) examples. We define a group to be logspace embeddable if it embeds in a group with normal forms computable in logspace. We prove that finitely generated nilpotent groups are logspace embeddable. It follows that all groups of polynomial growth are logspace embeddable.Comment: 24 pages, 1 figure. Minor corrections from previous versio

On conjugacy separability of fibre products

In this paper we study conjugacy separability of subdirect products of two free (or hyperbolic) groups. We establish necessary and sufficient criteria and apply them to fibre products to produce a finitely presented group $G_1$ in which all finite index subgroups are conjugacy separable, but which has an index $2$ overgroup that is not conjugacy separable. Conversely, we construct a finitely presented group $G_2$ which has a non-conjugacy separable subgroup of index $2$ such that every finite index normal overgroup of $G_2$ is conjugacy separable. The normality of the overgroup is essential in the last example, as such a group $G_2$ will always posses an index $3$ overgroup that is not conjugacy separable. Finally, we characterize $p$-conjugacy separable subdirect products of two free groups, where $p$ is a prime. We show that fibre products provide a natural correspondence between residually finite $p$-groups and $p$-conjugacy separable subdirect products of two non-abelian free groups. As a consequence, we deduce that the open question about the existence of an infinite finitely presented residually finite $p$-group is equivalent to the question about the existence of a finitely generated $p$-conjugacy separable full subdirect product of infinite index in the direct product of two free groups.Comment: v2: 38 pages; this is the version accepted for publicatio