arXiv subjects

Marco Linton

Publications and source records attributed to Marco Linton.

17 recordsLinked to original sources

arXiv2026

Finite complete rewriting systems for graphs of free groups with applications to free-by-cyclic, one-relator, and three-manifold groups

We prove that any finite graph of finitely generated free groups admits a finite complete rewriting system after possibly taking a free product with a free group of rank two. As a corollary we obtain that any HNN-extension of a finitely generated free group over a finitely generated subgroup admits a finite complete rewriting system. We then use this result, and other tools, to give partial solutions to several fundamental open problems about finite complete rewriting systems for hyperbolic, one-relator, fully residually free, and three-manifold groups. In particular we prove that if $G = \langle \mathbb{F} , t \mid t^{-1}ft = \psi(f), \, \forall f\in \mathbb{F} \rangle$ is the mapping torus of an injective endomorphism $\psi$ of a free group $\mathbb{F} $ (of possibly infinite rank) then every finitely generated subgroup of $G$ admits a finite complete rewriting system. It follows that any finitely generated virtually free-by-cyclic group, and any finitely generated subgroup of such a group, admits a finite complete rewriting system. We apply this to show that every finitely generated subgroup of a locally quasi-convex hyperbolic and virtually compact special group admits a finite complete rewriting system. This includes all one-relator groups with torsion (and all their finitely generated subgroups) and all hyperbolic fully residually free groups. Moreover, we show there is an algorithm that computes a finite complete rewriting system for any such group, given a presentation for its containing group and a finite list of generators for the subgroup. We also prove that for every compact three-manifold $M$, the group $\pi_1(M) \ast \mathbb{Z}$ admits a finite complete rewriting system. Furthermore, we show that the fundamental group of any compact three-manifold is autostackable and thus has a rational cross section and admits a bounded regular convergent prefix-rewriting system.

math.GR↗

arXiv2026

Coherent RFRS groups

We prove that a finitely generated virtually RFRS group of cohomological dimension at most $2$ is coherent if and only if its second $L^{2}$-Betti number vanishes if and only if it is virtually free-by-cyclic. The non-vanishing of the second $L^{2}$-Betti number provides the first known global obstruction to coherence in any reasonably wide class of groups, allowing for proofs of incoherence without needing to exhibit explicit witnesses to incoherence. As applications of this result, we completely characterise coherence among two-dimensional Coxeter groups, confirming conjectures of Jankiewicz and Wise, and show that incoherence is generic in groups of nonpositive deficiency, confirming a conjecture of Wise. We also find that, among virtually compact special groups of virtual cohomological dimension two, coherence is algorithmically decidable and is a quasi-isometry, measure equivalence, and profinite invariant. In an appendix, Marco Linton applies one of the main results to prove that cubulated locally quasi-convex hyperbolic groups are virtually free-by-cyclic, solving problems of Abdenbi--Wise and Wise in the cubulated case.

math.GR↗

arXiv2025

The finitely generated intersection property in fundamental groups of graphs of groups

A group $G$ is said to satisfy the finitely generated intersection property (f.g.i.p.) if the intersection of any two finitely generated subgroups of $G$ is again finitely generated. The aim of this article is to understand when the fundamental group of a graph of groups has the f.g.i.p. Our main results are general criteria for the f.g.i.p. in graphs of groups which depend on properties of the vertex groups, properties of certain double cosets of the edge groups and the structure of the underlying graph. For acylindrical graphs of groups, we also obtain criteria for the strong f.g.i.p. (s.f.g.i.p.). Our results generalise classical results due to Burns and Cohen on the f.g.i.p. for amalgamated free products and HNN extensions. As a concrete application, we show that a graph of locally quasi-convex hyperbolic groups with virtually $\mathbb{Z}$ edge groups (for instance, a generalised Baumslag--Solitar group) has the f.g.i.p. if and only if it does not contain $F_2\times\mathbb{Z}$ as a subgroup. In addition, we show that this condition is decidable. The main tools we use are the explicit constructions of pullbacks of immersions into a graph of group, obtained by the authors in a previous paper, and a technical condition on coset interactions, introduced in this paper.

math.GR↗

arXiv2025

Group pairs, coherence and Farrell--Jones Conjecture for $K_0$

A group pair $(G, X)$ consists of a group $G$ together with a $G$-set $X$. Such a pair encodes properties of $G$ relative to the stabilisers of points in $X$. In this paper, we show how to combine properties of group pairs and their stabilisers to prove coherence results for $G$ and its group algebra, as well as to study the quotient of $G$ obtained by killing the stabilisers. In particular, we prove that a torsion-free one-relator product of locally indicable groups is coherent provided that both factor groups are coherent. Moreover, we show that the group algebra of such a group over a field of characteristic $0$ is coherent whenever the group algebras of the factors are coherent. As other consequences of our methods, we also show that extensions of coherent locally indicable hyperbolic groups by $\mathbb{Z}$ are coherent and that groups admitting a Cohen--Lyndon presentation satisfy the Farrell--Jones Conjecture for $K_{0}$.

math.GR↗

arXiv2025

Embedding finitely generated free-by-cyclic groups in {finitely generated free}-by-cyclic groups

We refine Feighn--Handel's results on subgroups of mapping tori of free groups to the special case of free-by-cyclic groups. We use these refinements to show that any finitely generated free-by-cyclic group embeds in a {finitely generated free}-by-cyclic group. When the free-by-cyclic group is hyperbolic, it embeds in a hyperbolic {finitely generated free}-by-cyclic group as a quasi-convex subgroup. Combined with a result of Hagen--Wise, this implies that all hyperbolic free-by-cyclic groups are cocompactly cubulated.

math.GR↗

arXiv2025

The geometry of subgroups of mapping tori of free groups

We show that finitely generated mapping tori of free groups have a canonical collection of maximal sub-mapping tori of finitely generated free groups with respect to which they are relatively hyperbolic and locally relatively quasi-convex. As a consequence, we characterise locally quasi-convex hyperbolic groups amongst free-by-cyclic and one-relator groups. We also upgrade several known results for mapping tori of finitely generated free groups to the general case, such as the computations of Dehn functions, the solution to the conjugacy problem and the characterisation of the finitely generated intersection property.

math.GR↗

arXiv2025

Pullbacks and intersections in categories of graphs of groups

We develop a categorical framework for studying graphs of groups and their morphisms, with emphasis on pullbacks. More precisely, building on classical work by Serre and Bass, we give an explicit construction of the so-called $\mathbb{A}$-product of two morphisms into a graph of groups $\mathbb{A}$ -- a graph of groups which, within the appropriate categorical setting, captures the intersection of subgroups of the fundamental group of $\mathbb{A}$. We show that, in the category of pointed graphs of groups, pullbacks always exist and correspond precisely to pointed $\mathbb{A}$-products. In contrast, pullbacks do not always exist in the category of unpointed graphs of groups. However, when they do exist, and we show that it is the case, in particular, under certain acylindricity conditions, they are again closely related to $\mathbb{A}$-products. We trace, all along, the parallels with Stallings' classical theory of graph immersions and coverings, in relation to the study of the subgroups of free groups. Our results are useful for studying intersections of subgroups of groups that arise as fundamental groups of graphs of groups. As an example, we carry out an explicit computation of a pullback which results in a classification of the Baumslag--Solitar groups with the finitely generated intersection property.

math.GR↗

arXiv2025

The theory of one-relator groups: history and recent progress

The theory of one-relator groups is now almost a century old. The authors therefore feel that a comprehensive survey of this fascinating subject is in order, and this document is an attempt at precisely such a survey. This article is divided into two chapters, reflecting the two different phases in the story of one-relator groups. The first chapter, written by the second author, covers the historical development of the theory roughly until the advent of geometric group theory. The second chapter, written by the first author, covers the recent progress in the theory up until the present day. The two chapters can be read independently of one another and have minimal overlap.

math.GR↗

arXiv2024

Lifting relations in right orderable groups

In this article we study the following problem: given a chain complex $A_*$ of free $\mathbb{Z}G$-modules, when is $A_*$ isomorphic to the cellular chain complex of some simply connected $G$-CW-complex? Such a chain complex is called realisable. Wall studied this problem in the 60's and reduced it to a problem involving only the second differential $d_2$, now known as the relation lifting problem. We show that if $G$ is right orderable and $d_2$ is given by a matrix of a certain form, then $A_*$ is realisable. As a special case, we solve the relation lifting problem for right orderable groups with cyclic relation module.

math.GR↗

arXiv2024

Residually rationally solvable one-relator groups

We show that the intersection of the rational derived series of a one-relator group is rationally perfect and is normally generated by a single element. As a corollary, we characterise precisely when a one-relator group is residually rationally solvable.

math.GR↗

arXiv2023

Group rings of three-manifold groups

Let $G$ be the fundamental group of a three-manifold. By piecing together many known facts about three manifold groups, we establish two properties of the group ring $\mathbb{C}G$. We show that if $G$ has rational cohomological dimension two, then $\mathbb{C}G$ is coherent. We also show that if $G$ is torsion-free, then $G$ satisfies the Strong Atiyah Conjecture over $\mathbb{C}$ and hence that $\mathbb{C}G$ satisfies Kaplansky's Zero Divisor Conjecture.

math.GT↗

arXiv2023

On the coherence of one-relator groups and their group algebras

We prove that one-relator groups are coherent, solving a well-known problem of Gilbert Baumslag. Our proof strategy is readily applicable to many classes of groups of cohomological dimension two. We show that fundamental groups of two-complexes with non-positive immersions are homologically coherent, we show that groups with staggered presentations and many Coxeter groups are coherent and we show that group algebras over fields of characteristic zero of groups with reducible presentations without proper powers are coherent.

math.GR↗

arXiv2023

Virtually free-by-cyclic groups

We obtain a homological characterisation of virtually free-by-cyclic groups among groups that are hyperbolic and virtually compact special. As a consequence, we show that many groups known to be coherent actually possess the stronger property of being virtually free-by-cyclic. In particular, we show that all one-relator groups with torsion are virtually free-by-cyclic, solving a conjecture of Baumslag.

math.GR↗

arXiv2022

Hyperbolic one-relator groups

We introduce two families of two-generator one-relator groups called primitive extension groups and show that a one-relator group is hyperbolic if its primitive extension subgroups are hyperbolic. This reduces the problem of characterising hyperbolic one-relator groups to characterising hyperbolic primitive extension groups. These new groups moreover admit explicit decompositions as graphs of free groups with adjoined roots. In order to obtain this result, we characterise $2$-free one-relator groups with exceptional intersection in terms of Christoffel words, show that hyperbolic one-relator groups have quasi-convex Magnus subgroups and build upon the one-relator tower machinery developed in the authors previous article.

math.GR↗

arXiv2022

One-relator hierarchies

We prove that one-relator groups with negative immersions are hyperbolic and virtually special; this resolves a recent conjecture of Louder and Wilton. As a consequence, one-relator groups with negative immersions are residually finite, linear and have isomorphism problem decidable among one-relator groups. Using the fact that parafree one-relator groups have negative immersions, we answer a question of Baumslag's from 1986. The main new tool we develop is a refinement of the classic Magnus--Moldavanskii hierarchy for one-relator groups. We introduce the notions of Z-stable HNN-extensions and Z-stable hierarchies. We then show that a one-relator group is hyperbolic and has a quasi-convex one-relator hierarchy if and only if it does not contain a Baumslag--Solitar subgroup and has a Z-stable one-relator hierarchy.

math.GR↗

arXiv2021

The fully compressed subgroup membership problem

Suppose that $F$ is a free group and $k$ is a natural number. We show that the fully compressed membership problem for $k$-generated subgroups of $F$ is solvable in polynomial time. In order to do this, we adapt the theory of Stallings' foldings to handle edges with compressed labels. This partially answers a question of Markus Lohrey.

math.GR↗

arXiv2021

On the intersections of finitely generated subgroups of free groups: reduced rank to full rank

We show that the number of conjugacy classes of intersections $A\cap B^g$, for fixed finitely generated subgroups $A, B<F$ of a free group, is bounded above in terms of the ranks of $A$ and $B$; this confirms an intuition of Walter Neumann. This result was previously known only in the case where $A$ or $B$ is cyclic by the $w$-cycles theorem of Helfer and Wise and, independently, Louder and Wilton. In order to prove our main theorem, we introduce new results regarding the structure of fibre products of finite graphs. We work with a generalised definition of graph immersions so that our results apply to the theory of regular languages. For example, we give a new algorithm to decide non-emptiness of the intersection of two regular languages.

math.GR↗