The mapping torus of an endomorphism # of a group G is the HNNextension G#G with bonding maps the identity and #. We show that a mapping torus of an injective free group endomorphism has the property that its finitely generated subgroups are finitely presented and, moreover, these subgroups are of finite type. 1. Introduction A group is coherent if its finitely generated subgroups are finitely presented. Free groups are obviously coherent. The classification of surfaces and the fact that every cover of a surface is itself a surface, imply that surface groups are coherent. In the early 1970's, Scott [Sco73] and Shalen (unpublished) independently answered a question of Jaco by showing that the fundamental group of a 3-manifold is coherent. Stallings [Sta77] showed that F 2 F 2 is not coherent. Since most finitely generated groups are not finitely presented, the question of which groups are coherent has centered on groups with special properties. Rips [Rip82] gave examples of incohere...

Abstract. A group is coherent if all its finitely generated subgroups are finitely presented. In this article we provide a criterion for positively determining the coherence of a group. This criterion is based upon the notion of the perimeter of a map between two finite 2-complexes which is introduced here. In the groups to which this theory applies, a presentation for a finitely generated subgroup can be computed in quadratic time relative to the sum of the lengths of the generators. For many of these groups we can show in addition that they are locally quasiconvex. As an application of these results we prove that one-relator groups with sufficient torsion are coherent and locally quasiconvex and we give an alternative proof of the coherence and local quasiconvexity of certain 3-manifold groups. The main application is to establish the coherence

Given a finitely generated (fg) group G, the set R(G) of homomorphisms from G to SL2C inherits the structure of an algebraic variety known as the representation variety of G in SL2C. This algebraic variety is an invariant of fg presentations of G. Call a group G parafree of rank n if it shares the lower central sequence with a free group of rank n, and if it is residually nilpotent. The deviation of a fg parafree group is the difference between the minimum possible number of generators of G and the rank of G. So parafree groups of deviation zero are actually just free groups. Parafree groups that are not free share a host of properties with free groups. In this paper algebraic geometric invariants involving the number of maximal irreducible components (mirc) of R(G), and the dimension of R(G) for certain classes of parafree groups are computed. It is shown that in an infinite number of cases these invariants successfully discriminate between isomorphism types within the class of parafree groups of the same rank. This is quite surprising, since an n generated group G is free of rank n iff Dim(R(G)) = 3n. In fact, a direct consequence of Theorem 1.6 in this paper is that given an arbitrary positive integer k, and any integer r ≥ 2 there exist infinitely many non-isomorphic fg parafree groups of rank r and deviation one with representation varieties of dimension 3r, having more than k mirc of dimension 3r. This paper also introduces many novel and surprising dimension formulas for the representation varieties of certain one-relator groups. General Structure of the Paper. This paper begins with an introduction where relevant ideas to what will follow are developed. It then goes on to define what a parafree group is, and how the notions that inspired G. Baumslag to give rise to such groups arose in the context of investigations conducted by W. Magnus, and a question of Hanna Neumann. The new results in this paper are Theorem 1.0, Theorem 1.1, Theorem 1.2, Corollary 1.2, Theorem 1.3, Theorem 1.5, and Theorem 1.6. In Section One several results from the author’s earlier work are introduced. In Section Two the following theorems are proven: 1.0, 1.1, and 1.6. The paper ends with a list notation.

This paper introduces and surveys the theory of Malcev presentations. [Malcev presentations are a species of presentation that can be used to define any semigroup that can be embedded into a group.] In particular, various classes of groups and monoids all of whose finitely generated subsemigroups admit finite Malcev presentations are described; closure and containment results are stated; links with the theory of automatic semigroups are mentioned; and various questions asked. Many of the results stated herein are summarized in tabular form. 1

Abstract. We prove that if G is a group that splits as a finite graph of finitely generated free groups with cyclic edge groups, and G has no non-Euclidean Baumslag-Solitar subgroups, then G is the fundamental group of a compact nonpositively curved cube complex. In addition, if G is also wordhyperbolic (i.e., if G contains no Baumslag-Solitar subgroups of any type), we show that G is linear (in fact, is a subgroup of SLn(Z)). Contents.

Abstract. A well-known conjecture says that every one-relator group is coherent. We state and partly prove a similar statement for graded associative algebras. In particular, we show that every Gorenstein algebra A of global dimension 2 is graded coherent. This allows us to define a noncommutative analogue of the projective line P1 as a noncommutative scheme based on the coherent noncommutative spectrum

Abstract. Hanna Neumann asked whether it was possible for two non-isomorphic residually nilpotent groups, one of them free, to share the lower central sequence. G. Baumslag answered the question in the affirmative and thus gave rise to parafree groups. A group G is termed parafree of rank n if it is residually nilpotent and shares the same lower central sequence with a free group of rank n. The deviation of a fg parafree group G is the difference µ(G) − µ ( G), where µ(G) is the minimum γ2G possible number of generators of G, and γ2G is the second term of the lower central series of G. Let G be a finitely generated group (fg), then Hom(G, SL(2, C)) inherits the structure of an algebraic variety, denoted by R(G), which is an invariant of fg presentations of G. If G is an n generated parafree group, then the deviation of G is 0 iff Dim(R(G)) = 3n. It is known that for n ≥ 2 there exist infinitely many parafree groups of rank n and deviation 1 with non-isomorphic representation varieties of dimension 3n. In this paper it is shown that given integers n ≥ 2, and k ≥ 1, there exists infinitely many parafree groups of rank n and deviation k with non-isomorphic representation varieties of dimension different from 3n; in particular, there exist infinitely many parafree groups G of rank n with Dim(R(G))> q, where q ≥ 3n is an arbitrary integer.

Mapping tori of free group automorphisms are coherent By Mark Feighn and Michael Handel* The mapping torus of an endomorphism Φ of a group G is the HNNextension G∗G with bonding maps the identity and Φ. We show that a mapping torus of an injective free group endomorphism has the property that its finitely generated subgroups are finitely presented and, moreover, these subgroups are of finite type. 1.