All Fuchsian Schottky groups are classical Schottky groups

Not all Schottky groups of Moebius transformations are classical Schottky groups. In this paper we show that all Fuchsian Schottky groups are classical Schottky groups, but not necessarily on the same set of generators.Comment: 9 pages. Published copy, also available at http://www.maths.warwick.ac.uk/gt/GTMon1/paper5.abs.htm

Grades of Discrimination: Indiscernibility, symmetry, and relativity

There are several relations which may fall short of genuine identity, but which behave like identity in important respects. Such grades of discrimination have recently been the subject of much philosophical and technical discussion. This paper aims to complete their technical investigation. Grades of indiscernibility are defined in terms of satisfaction of certain first-order formulas. Grades of symmetry are defined in terms of symmetries on a structure. Both of these families of grades of discrimination have been studied in some detail. However, this paper also introduces grades of relativity, defined in terms of relativeness correspondences. This paper explores the relationships between all the grades of discrimination, exhaustively answering several natural questions that have so far received only partial answers. It also establishes which grades can be captured in terms of satisfaction of object-language formulas, and draws connections with definability theory.Comment: Minor changes: a table has been added to section 2 (for user reference), and the identity-free version of Beth-Svenonius in section 6 gets a slightly nicer treatmen

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.Comment: 37 pages including 6 pages of table

Strictly ascending HNN extensions of finite rank free groups that are linear over Z

We find strictly ascending HNN extensions of finite rank free groups possessing a presentation 2-complex which is a non positively curved square complex. On showing these groups are word hyperbolic, we have by results of Wise and Agol that they are linear over the integers. An example is the endomorphism of the free group on a,b with inverses A,B that sends a to aBaab and b to bAbba.Comment: 21 pages, just 1 figur

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.Comment: 29 page

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.Comment: 10 page

Balanced groups and graphs of groups with infinite cyclic edge groups

We give a necessary and sufficient condition for the fundamental group of a finite graph of groups with infinite cyclic edge groups to be acylindrically hyperbolic, from which it follows that a finitely generated group splitting over Z cannot be simple. We also give a necessary and sufficient condition (when the vertex groups are torsion free) for the fundamental group to be balanced, where a group is said to be balanced if $x^m$ conjugate to $x^n$ implies that $|m|=|n|$ for all infinite order elements $x$

Acylindrical hyperbolicity, non simplicity and SQ-universality of groups splitting over Z

We show, using acylindrical hyperbolicity, that a finitely generated group splitting over $\Z$ cannot be simple. We also obtain SQ-universality in most cases, for instance a balanced group (one where if two powers of an infinite order element are conjugate then they are equal or inverse) which is finitely generated and splits over $\Z$ must either be SQ-universal or it is one of exactly seven virtually abelian exceptions.Comment: Much shorter version of 1509.05688 with strengthening of main resul

Groups possessing only indiscrete embeddings in SL(2,C)

We give results on when a finitely generated group has only indiscrete embeddings in SL(2,C), with particular reference to 3-manifold groups. For instance if we glue two copies of the figure 8 knot along its torus boundary then the fundamental group of the resulting closed 3-manifold sometimes embeds in SL(2,C) and sometimes does not, depending on the identification. We also give another quick counterexample to Minsky's simple loop question.Comment: Minor changes and update

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