arXiv subjects
Publications and source records attributed to James Howie.
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.
Given an abelian group $A$ and a Lie group $G$, we construct a bilinear pairing from $A\timesπ_1({\mathcal R})$ to $π_1(G)$, where $\mathcal R$ is a subvariety of the variety of representations $A\to G$. In the case where $A$ is the peripheral subgroup of a torus or two-bridge knot group, $G=S^1$ and $\mathcal R$ is a certain variety of representations arising from suitable SU(2)-representations of the knot group, we show that this pairing is not identically zero. We discuss the consequences of this result for the SU(2)-representations of fundamental groups of manifolds obtained by Dehn surgery on such knots.
A Magnus subgroup of a one-relator group is the free subgroup freely generated by a proper subset of the generators. Two such subgroups can intersect in the obvious way or in a larger, exceptional way. The condition of non-exceptional intersection of Magnus subgroups of a one-relator group is applied to give criteria for the resulting cyclically presented groups to contain nonabelian free subgroups.
A generalized triangle group is a group that can be presented in the form $G = < x,y | x^p=y^q=w(x,y)^r=1>$, where $p,q,r\geq 2$ and $w(x,y)$ is a cyclically reduced word of length at least 2 in the free product $\Z_p*\Z_q=< x,y | x^p=y^q=1>$. Rosenberger has conjectured that every generalized triangle group $G$ satisfies the Tits alternative. It is known that the conjecture holds except possibly when the triple $(p,q,r)$ is one of $(2,3,2), (2,4,2),(2,5,2),(3,3,2),(3,4,2)$, or $(3,5,2)$. In this paper we show that the Tits alternative holds in the case $(p,q,r)=(3,4,2)$.
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.
We exploit Zlil Sela's description of the structure of groups having the same elementary theory as free groups: they and their finitely generated subgroups form a prescribed subclass E of the hyperbolic limit groups. We prove that if $G_1,...,G_n$ are in E then a subgroup $Γ\subset G_1\times...\times G_n$ is of type $\FP_n$ if and only if $Γ$ is itself, up to finite index, the direct product of at most $n$ groups from $\mathcal E$. This answers a question of Sela.
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.
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.