arXiv subjects

J. O. Button

Publications and source records attributed to J. O. Button.

28 records · Page 2Linked to original sources

arXiv2010

Virtual finite quotients of finitely generated groups

If G is a semidirect product N by H with N normal and finitely generated then G has the property that every finite group is a quotient of some finite index subgroup of G if and only if one of N and H has this property. This has applications to 3-manifolds and to cyclically presented groups, for instance for any fibred hyperbolic 3-manifold M and any finite simple group S, there is a cyclic cover of M whose fundamental group surjects to S. We also give a short proof of the residual finiteness of ascending HNN extensions of finite rank free groups when the induced map on homology is injective.

math.GR↗

arXiv2010

Applications of p-deficiency and p-largeness

We use Schlage-Puchta's concept of p-deficiency and Lackenby's property of p-largeness to show that a group having a finite presentation with p-deficiency greater than 1 is large, which implies that Schlage-Puchta's infinite finitely generated p-groups are not finitely presented. We also show that for all primes p at least 7, any group having a presentation of p-deficiency greater than 1 is Golod-Shafarevich, and has a finite index subgroup which is Golod-Shafarevich for the remaining primes. We also generalise a result of Grigorchuk on Coxeter groups to odd primes.

math.GR↗

arXiv2010

Strictly ascending HNN extensions in soluble groups

We show that there exist finitely generated soluble groups which are not LERF but which do not contain strictly ascending HNN extensions of a cyclic group. This solves Problem 16.2 in the Kourovka notebook. We further show that there is a finitely presented soluble group which is not LERF but which does not contain a strictly ascending HNN extension of a polycyclic group.

math.GR↗

arXiv2009

Non proper HNN extensions and uniform uniform exponential growth

If a finitely generated torsion free group K has the property that all finitely generated subgroups S of K are either small or have growth constant bounded uniformly away from 1 then a non proper HNN extension G of K, that is a semidirect product of K by the integers, has the same property. Here small means cyclic or, if the automorphism has no periodic conjugacy classes, free abelian of bounded rank.

math.GR↗

arXiv2008

Proving finitely presented groups are large by computer

We present a theoretical algorithm which, given any finite presentation of a group as input, will terminate with answer yes if and only if the group is large. We then implement a practical version of this algorithm using Magma and apply it to a range of presentations. Our main focus is on 2-generator 1-relator presentations where we have a complete picture of largeness if the relator has exponent sum zero in one generator and word length at most 12, as well as when the relator is in the commutator subgroup and has word length at most 18. Indeed all but a tiny number of presentations define large groups. Finally we look at fundamental groups of closed hyperbolic 3-manifolds, where the algorithm readily determines that a quarter of the groups in the Snappea closed census are large.

math.GR↗

arXiv2007

Large Groups of Deficiency One

We prove that if a group possesses a deficiency 1 presentation where one of the relators is a commutator then it is the integers times the integers, is large, or is as far as possible from being residually finite. Then we use this to show that a mapping torus of an endomorphism of a finitely generated free group is large if it contains the integers times the integers as a subgroup of infinite index, as well as showing that such a group is large if it contains a Baumslag-Solitar group of infinite index and has a finite index subgroup with first Betti number at least 2. We give applications to free by cyclic groups, 1 relator groups and residually finite groups.

math.GR↗

arXiv2007

A formula for the normal subgroup growth of Baumslag-Solitar groups

We give an exact formula for the number of normal subgroups of each finite index in the Baumslag-Solitar group BS(p,q) when p and q are coprime. Unlike the formula for all finite index subgroups, this one distinguishes different Baumslag-Solitar groups and is not multiplicative. This allows us to give an example of a finitely generated profinite group which is not virtually pronilpotent but whose zeta function has an Euler product.

math.GR↗

arXiv2007

Finite covers of the infinite cyclic cover of a knot

We show that the commutator subgroup G' of a classical knot group G need not have subgroups of every finite index, but it will if G' has a surjective homomorphism to the integers and we give an exact criterion for that to happen. We also give an example of a smoothly knotted n-sphere in the (n+2)-sphere for all n at least 2 whose infinite cyclic cover is not simply connected but has no proper finite covers.

math.GT↗

arXiv2005

Large mapping tori of free group endomorphisms

We present an algorithm which, given any finite presentation of a group as input, will terminate with answer yes if and only if the group is large. We use this to prove that a mapping torus of a finitely generated free group automorphism is large if it contains the integers times the integers as a subgroup of infinite index. We then extend this result to mapping tori of finitely generated free group endomorphisms, as well as showing that such a group is large if it contains a Baumslag-Solitar group of infinite index and has a finite index subgroup with first Betti number at least 2. We also show that if a group possesses a deficiency 1 presentation where one of the relators is a commutator then it is the integers times the integers, or it is large, or it is as far as possible from being residually finite.

math.GR↗

arXiv2004

Fibred and Virtually Fibred hyperbolic 3-manifolds in the censuses

Following on from work of Dunfield, we determine the fibred status of all the unknown hyperbolic 3-manifolds in the cusped census. We then find all the fibred hyperbolic 3-manifolds in the closed census and use this to find over 100 examples each of closed and cusped virtually fibred non-fibred census 3-manifolds, including the Weeks manifold. We also show that the co-rank of the fundamental group of every 3-manifold in the cusped and in the closed census is 0 or 1.

math.GT↗