Minimal Seifert manifolds for higher ribbon knots

We show that a group presented by a labelled oriented tree presentation in which the tree has diameter at most three is an HNN extension of a finitely presented group. From results of Silver, it then follows that the corresponding higher dimensional ribbon knots admit minimal Seifert manifolds.Comment: 33 pages. Published copy, also available at http://www.maths.warwick.ac.uk/gt/GTMon1/paper12.abs.htm

Can Dehn surgery yield three connected summands?

A consequence of the Cabling Conjecture of Gonzalez-Acu\~{n}a and Short is that Dehn surgery on a knot in $S^3$ cannot produce a manifold with more than two connected summands. In the event that some Dehn surgery produces a manifold with three or more connected summands, then the surgery parameter is bounded in terms of the bridge number by a result of Sayari. Here this bound is sharpened, providing further evidence in favour of the Cabling Conjecture.Comment: 11 pages, 2 figure

Non-triviality of some one-relator products of three groups

In this paper we study a group G which is the quotient of a free product of three non-trivial groups by the normal closure of a single element. In particular we show that if the relator has length at most eight, then G is non-trivial. In the case where the factors are cyclic, we prove the stronger result that at least one of the factors embeds in G.Comment: 21 pages, 3 figure

Subgroups of direct products of two limit groups

If S is a subgroup of a direct product of two limit groups, and S is of type FP(2) over the rationals, then S has a subgroup of finite index that is a direct product of at most two limit groups.Comment: 18 pages, no figure

Freiheitss\"{a}tze for one-relator quotients of surface groups and of limit groups

Three versions of the Freiheitssatz are proved in the context of one-relator quotients of limit groups, where the latter are equipped with 1-acylindrical splittings over cyclic subgroups. These are natural extensions of previously published corresponding statements for one-relator quotients of orientable surface groups. Two of the proofs are new even in that restricted context.Comment: 17 page

Normalisers in Limit Groups

Let \G be a limit group, S\subset\G a subgroup, and $N$ the normaliser of $S$. If $H_1(S,\mathbb Q)$ has finite \Q-dimension, then $S$ is finitely generated and either $N/S$ is finite or $N$ is abelian. This result has applications to the study of subdirect products of limit groups.Comment: 10 pages, no figure