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Henry Wilton

Publications and source records attributed to Henry Wilton.

At least 19 recordsLinked to original sources
arXiv2025

Power quotients of surface groups and mapping class groups

Let $\Gamma$ be the fundamental group of a closed, orientable, hyperbolic surface $S$. The $n$-power quotient, $\Gamma(n)$, is the quotient of $\Gamma$ by the $n$th powers of simple closed curves. We prove an analogue of the Dehn--Nielsen--Baer theorem for suitable large values of $n$: the outer automorphism group of $\Gamma(n)$ is isomorphic to the quotient of the extended mapping class group of $S$ by $n$th powers of Dehn twists. There is also a corresponding description of the automorphism group as the quotient of the extended mapping class group of the corresponding once-punctured surface, and we relate these groups via a Birman-type exact sequence. Along the way, and as consequences, we prove structural properties of $\Gamma(n)$ for suitable large values of $n$, including: $\Gamma(n)$ is virtually torsion-free, acylindrically hyperbolic, infinitely presented, with solvable word problem and finite asymptotic dimension.

math.GR↗
arXiv2024

Surface groups among cubulated hyperbolic and one-relator groups

Let $X$ be a non-positively curved cube complex with hyperbolic fundamental group. We prove that $\pi_1(X)$ has a non-free subgroup of infinite index unless $\pi_1(X)$ is either free or a surface group, answering questions of Gromov and Whyte (in a special case) and Wise. A similar result for one-relator groups follows, answering a question posed by several authors. The proof relies on a careful analysis of free and cyclic splittings of cubulated groups.

math.GR↗
arXiv2022

Rationality theorems for curvature invariants of 2-complexes

Let $X$ be a finite, 2-dimensional cell complex. The curvature invariants $\rho_\pm(X)$ and $\sigma_\pm(X)$ were defined in [13], and a programme of conjectures was outlined. Here, we prove the foundational result that the quantities $\rho_\pm(X)$ and $\sigma_\pm(X)$ are the extrema of explicit rational linear-programming problems. As a result they are rational, realised, and can be computed algorithmically.

math.GR↗
arXiv2022

Rational curvature invariants for 2-complexes

New invariants for 2-dimensional cell complexes are defined, which can be interpreted as curvature bounds. These invariants are proved to be rational and computable in a companion article. This document is a survey that collects theorems about these invariants, computes examples, and lays out a programme of conjectures.

math.GR↗
arXiv2021

Uniform negative immersions and the coherence of one-relator groups

Previously, the authors proved that the presentation complex of a one-relator group $G$ satisfies a geometric condition called negative immersions if every two-generator, one-relator subgroup of $G$ is free. Here, we prove that one-relator groups with negative immersions are coherent, answering a question of Baumslag in this case. Other strong constraints on the finitely generated subgroups also follow such as, for example, the co-Hopf property. The main new theorem strengthens negative immersions to uniform negative immersions, using a rationality theorem proved with linear-programming techniques.

math.GR↗
arXiv2021

Negative immersions for one-relator groups

We prove a freeness theorem for low-rank subgroups of one-relator groups. Let $F$ be a free group, and let $w\in F$ be a non-primitive element. The primitivity rank of $w$, $π(w)$, is the smallest rank of a subgroup of $F$ containing $w$ as an imprimitive element. Then any subgroup of the one-relator group $G=F/\langle\langle w\rangle\rangle$ generated by fewer than $π(w)$ elements is free. In particular, if $π(w)>2$ then $G$ doesn't contain any Baumslag--Solitar groups. The hypothesis that $π(w)>2$ implies that the presentation complex $X$ of the one-relator group $G$ has negative immersions: if a compact, connected complex $Y$ immerses into $X$ and $χ(Y)\geq 0$ then $Y$ is Nielsen equivalent to a graph. The freeness theorem is a consequence of a dependence theorem for free groups, which implies several classical facts about free and one-relator groups, including Magnus' Freiheitssatz and theorems of Lyndon, Baumslag, Stallings and Duncan--Howie. The dependence theorem strengthens Wise's $w$-cycles conjecture, proved independently by the authors and Helfer--Wise, which implies that the one-relator complex $X$ has non-positive immersions when $π(w)>1$.

math.GR↗
arXiv2020

One-relator groups with torsion are coherent

We show that any one-relator group $G=F/\langle\langle w\rangle\rangle$ with torsion is coherent -- i.e., that every finitely generated subgroup of $G$ is finitely presented -- answering a 1974 question of Baumslag in this case.

math.GR↗
arXiv2018

Profinite detection of 3-manifold decompositions

The profinite completion of the fundamental group of a closed, orientable $3$-manifold determines the Kneser--Milnor decomposition. If $M$ is irreducible, then the profinite completion determines the Jaco--Shalen--Johannson decomposition of $M$.

math.GR↗
arXiv2018

Generalized triangle groups, expanders, and a problem of Agol and Wise

Answering a question asked by Agol and Wise, we show that a desired stronger form of Wise's malnormal special quotient theorem does not hold. The counterexamples are generalizations of triangle groups, built using the Ramanujan graphs constructed by Lubotzky--Phillips--Sarnak.

math.GR↗
arXiv2018

Essential surfaces in graph pairs

A well known question of Gromov asks whether every one-ended hyperbolic group $Γ$ has a surface subgroup. We give a positive answer when $Γ$ is the fundamental group of a graph of free groups with cyclic edge groups. As a result, Gromov's question is reduced (modulo a technical assumption on 2-torsion) to the case when $Γ$ is rigid. We also find surface subgroups in limit groups. It follows that a limit group with the same profinite completion as a free group must in fact be free, which answers a question of Remeslennikov in this case.

math.GR↗
arXiv2017

Profinite rigidity and surface bundles over the circle

If $M$ is a compact 3-manifold whose first betti number is 1, and $N$ is a compact 3-manifold such that $π_1N$ and $π_1M$ have the same finite quotients, then $M$ fibres over the circle if and only if $N$ does. We prove that groups of the form $F_2\rtimes\mathbb{Z}$ are distinguished from one another by their profinite completions. Thus, regardless of betti number, if $M$ and $N$ are punctured torus bundles over the circle and $M$ is not homeomorphic to $N$, then there is a finite group $G$ such that one of $π_1M$ and $π_1N$ maps onto $G$ and the other does not.

math.GR↗
arXiv2017

Pro-$p$ subgroups of profinite completions of 3-manifold groups

We completely describe the finitely generated pro-$p$ subgroups of the profinite completion of the fundamental group of an arbitrary $3$-manifold. We also prove a pro-$p$ analogue of the main theorem of Bass--Serre theory for finitely generated pro-$p$ groups.

math.GR↗
arXiv2017

On the recognition problem for virtually special cube complexes

We address the question of whether the property of being virtually special (in the sense of Haglund and Wise) is algorithmically decidable for finite, non-positively curved cube complexes. Our main theorem shows that it cannot be decided locally, i.e. by examining one hyperplane at a time. Specifically, we prove that there does not exist an algorithm that, given a compact non-positively squared 2-complex X and a hyperplane H in X can decide whether or not there is a finite-sheeted cover of X in which no lift of H self-osculates.

math.GR↗
arXiv2017

The structure of limit groups over hyperbolic groups

Let $Γ$ be a torsion-free hyperbolic group. We study $Γ$--limit groups which, unlike the fundamental case in which $Γ$ is free, may not be finitely presentable or geometrically tractable. We define model $Γ$--limit groups, which always have good geometric properties (in particular, they are always relatively hyperbolic). Given a strict resolution of an arbitrary $Γ$--limit group $L$, we canonically construct a strict resolution of a model $Γ$--limit group, which encodes all homomorphisms $L\to Γ$ that factor through the given resolution. We propose this as the correct framework in which to study $Γ$--limit groups algorithmically. We enumerate all $Γ$--limit groups in this framework.

math.GR↗
arXiv2017

Immutability is not uniformly decidable in hyperbolic groups

A finitely generated subgroup H of a torsion-free hyperbolic group G is called immutable if there are only finitely many conjugacy classes of injections of H into G. We show that there is no uniform algorithm to recognize immutability, answering a uniform version of a question asked by the authors.

math.GR↗
arXiv2017

3-manifolds everywhere

A random group contains many subgroups which are isomorphic to the fundamental group of a compact hyperbolic 3-manifold with totally geodesic boundary. These subgroups can be taken to be quasi-isometrically embedded. This is true both in the few relators model, and the density model of random groups (at any density less than a half).

math.GR↗