The homeomorphism problem for closed 3-manifolds

We give a more geometric approach to an algorithm for deciding whether two hyperbolic 3-manifolds are homeomorphic. We also give a more algebraic approach to the homeomorphism problem for geometric, but non-hyperbolic, 3-manifolds.Comment: first version: 12 pages. Replacement: 14 pages. Includes minor improvements to exposition in response to referee's comment

GRAPH SMALL CANCELLATION THEORY APPLIED TO ALTERNATING LINK GROUPS

International audienceWe show that the Wirtinger presentation of a prime alternating link group satisfies a generalized small cancellation theory condition. This gives a simplification of Weinbaum’s solution to the word and conjugacy problems for these groups

Generalising Collins' Theorem

We generalise a result of D. J. Collins on intersections of conjugates of Magnus subgroups of one-relator groups to the context of one-relator products of locally indicable groups.Comment: 28 pages, 3 figure

INVERSION IS POSSIBLE IN GROUPS WITH NO PERIODIC AUTOMORPHISMS

International audienceThere exist infinite, finitely presented, torsion-free groups G such that Aut(G) and Out(G) are torsion-free but G has an automorphism sending some non-trivial element to its inverse

On the finite presentation of subdirect products and the nature of residually free groups

We establish {\em{virtual surjection to pairs}} (VSP) as a general criterion for the finite presentability of subdirect products of groups: if $\Gamma_1,...,\Gamma_n$ are finitely presented and $S<\Gamma_1\times...\times\Gamma_n$ projects to a subgroup of finite index in each $\Gamma_i\times\Gamma_j$, then $S$ is finitely presentable, indeed there is an algorithm that will construct a finite presentation for $S$. We use the VSP criterion to characterise the finitely presented residually free groups. We prove that the class of such groups is recursively enumerable. We describe an algorithm that, given a finite presentation of a residually free group, constructs a canonical embedding into a direct product of finitely many limit groups. We solve the (multiple) conjugacy problem and membership problem for finitely presentable subgroups of residually free groups. We also prove that there is an algorithm that, given a finite generating set for such a subgroup, will construct a finite presentation. New families of subdirect products of free groups are constructed, including the first examples of finitely presented subgroups that are neither ${\rm{FP}}_\infty$ nor of Stallings-Bieri typeComment: 44 pages. To appear in American Journal of Mathematics. This is a substantial rewrite of our previous Arxiv article 0809.3704, taking into account subsequent developments, advice of colleagues and referee's comment

Finitely presented subgroups of automatic groups and their isoperimetric functions

We describe a general technique for embedding certain amalgamated products into direct products. This technique provides us with a way of constructing a host of finitely presented subgroups of automatic groups which are not even asynchronously automatic. We can also arrange that such subgroups satisfy, at best, an exponential isoperimetric inequality.Comment: DVI and Post-Script files only. To appear in J. London Math. So

Subgroups of direct products of limit groups

If $G_1,...,G_n$ are limit groups and $S\subset G_1\times...\times G_n$ is of type \FP_n(\mathbb Q) then $S$ contains a subgroup of finite index that is itself a direct product of at most $n$ limit groups. This settles a question of Sela.Comment: 20 pages, no figures. Final version. Accepted by the Annals of Mathematic

Topological methods in group theory : the adjunction problem

The work presented here is a new method of attack on an old group theory problem known as the Adjunction Problem, defined by B H Neumann in 19^3 (see [Nj ). The problem is the following : given a group, form a new group by adding one new generator and one new relation ; determine the conditions under which the natural map from the original to the modified group is an injection. (For instance the new relator must not be conjugate to a word in the original group.) The main result obtained using the new methods is that the map is indeed an injection when the original group is locally indicable - a new result independently obtained by Howie [HJ and Brodskii [Brj . Chapter 1 consists of some basic definitions and some of the known results, together with statements of the new results and some instances of where the problem arises in lowdimensional topology. In Chapter 2 we introduce the new methods - showing that a non-trivial element in the kernal of the natural map for a given group and added relator (a "counter-example") gives us a labelled, planar graph with certain properties (a "special diagram") and that this special diagram in its turn defines a counter-example (these results are summed up in 2.22). These topologically obtained diagrams turn out (2.10) to be dual to the "Dehn diagrams" of Small Cancellation Theory (see for instance [Ls] or l_L2J ). In Chapter 3 a class of such diagrams is constructed and it is shown that none of these corresponds to a counterexample. This class contains the only diagrams known to the author which give potential counter-examples ("triples") such that the new generator appears with exponent-sum non-zero in the added relator. Chapter *+ begins with the construction of a potential function on a diagram, based on work by Lyndon [L2J . This is then used to prove the main result of the thesis, the Freiheits- satz for locally indicable groups, a new proof of the result which (as noted above) has been independently obtained by Howie and by Brodskii. Finally it is show that the existence of a counter-example for a given group G and a given relator r depends upon the existence of a counter-example for G* and added relator r" , where r" is one of two words obtained from r using a homomorphism from G-* to 71* which takes r to zero