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Henry Wilton

Publications and source records attributed to Henry Wilton.

At least 37 records · Page 2Linked to original sources
arXiv2017

Undecidability and the developability of permutoids and rigid pseudogroups

A permutoid is a set of partial permutations that contains the identity and is such that partial compositions, when defined, have at most one extension in the set. In 2004 Peter Cameron conjectured that there can exist no algorithm that determines whether or not a finite permutoid based on a finite set can be completed to a finite permutation group, and he related this problem to the study of groups that have no non-trivial finite quotients. This note explains how our recent work on the profinite triviality problem for finitely presented groups can be used to prove Cameron's conjecture. We also prove that the existence problem for finite developments of rigid pseudogroups is unsolvable.

math.GR↗
arXiv2016

Distinguishing geometries using finite quotients

We prove that the profinite completion of the fundamental group of a compact 3-manifold $M$ satisfies a Tits alternative: if a closed subgroup $H$ does not contain a free pro-$p$ subgroup for any $p$, then $H$ is virtually soluble, and furthermore of a very particular form. In particular, the profinite completion of the fundamental group of a closed, hyperbolic 3-manifold does not contain a subgroup isomorphic to $\hat{\mathbb{Z}}^2$. This gives a profinite characterization of hyperbolicity among irreducible 3-manifolds. We also characterize Seifert fibred 3-manifolds as precisely those for which the profinite completion of the fundamental group has a non-trivial procyclic normal subgroup. Our techniques also apply to hyperbolic, virtually special groups, in the sense of Haglund and Wise. Finally, we prove that every finitely generated pro-$p$ subgroup of the profinite completion of a torsion-free, hyperbolic, virtually special group is free pro-$p$.

math.GR↗
arXiv2015

Decision problems for 3-manifolds and their fundamental groups

We survey the status of some decision problems for 3-manifolds and their fundamental groups. This includes the classical decision problems for finitely presented groups (Word Problem, Conjugacy Problem, Isomorphism Problem), and also the Homeomorphism Problem for 3-manifolds and the Membership Problem for 3-manifold groups.

math.GT↗
arXiv2015

The triviality problem for profinite completions

We prove that there is no algorithm that can determine whether or not a finitely presented group has a non-trivial finite quotient; indeed, this remains undecidable among the fundamental groups of compact, non-positively curved square complexes. We deduce that many other properties of groups are undecidable. For hyperbolic groups, there cannot exist algorithms to determine largeness, the existence of a linear representation with infinite image (over any infinite field), or the rank of the profinite completion.

math.GR↗
arXiv2014

Stackings and the W-cycle conjecture

We prove Wise's $W$-cycles conjecture. Consider a compact graph $Γ'$ immering into another graph $Γ$. For any immersed cycle $Λ:S^1\to Γ$, we consider the map $Λ'$ from the circular components $\mathbb{S}$ of the pullback to $Γ'$. Unless $Λ'$ is reducible, the degree of the covering map $\mathbb{S}\to S^1$ is bounded above by minus the Euler characteristic of $Γ'$. As a consequence, we obtain a homological version of coherence for one-relator groups.

math.GR↗
arXiv2014

The isomorphism problem for profinite completions of residually finite groups

We consider pairs of finitely presented, residually finite groups $u:P\hookrightarrow Γ$. We prove that there is no algorithm that, given an arbitrary such pair, can determine whether or not the associated map of profinite completions $\hat{u}: \widehat{P} \to \widehatΓ$ is an isomorphism. Nor do there exist algorithms that can decide whether $\hat{u}$ is surjective, or whether $\widehat{P}$ is isomorphic to $\widehatΓ$.

math.GR↗
arXiv2013

3-manifold groups

We summarize properties of 3-manifold groups, with a particular focus on the consequences of the recent results of Ian Agol, Jeremy Kahn, Vladimir Markovic and Dani Wise.

math.GT↗
arXiv2012

Recognizing geometric 3-manifold groups using the word problem

Adyan and Rabin showed that most properties of groups cannot be algorithmically recognized from a finite presentation alone. We prove that, if one is also given a solution to the word problem, then the class of fundamental groups of closed, geometric 3-manifolds is algorithmically recognizable. In our terminology, the class of geometric 3-manifold groups is `recursive modulo the word problem'.

math.GR↗
arXiv2012

Profinite properties of graph manifolds

Let $M$ be a closed, orientable, irreducible, geometrizable 3-manifold. We prove that the profinite topology on the fundamental group of $π_1(M)$ is efficient with respect to the JSJ decomposition of $M$. We go on to prove that $π_1(M)$ is good, in the sense of Serre, if all the pieces of the JSJ decomposition are. We also prove that if $M$ is a graph manifold then $π_1(M)$ is conjugacy separable.

math.GR↗
arXiv2012

Separability of double cosets and conjugacy classes in 3-manifold groups

Let M = H^3 / Γbe a hyperbolic 3-manifold of finite volume. We show that if H and K are abelian subgroups of Γand g is in Γ, then the double coset HgK is separable in Γ. As a consequence we prove that if M is a closed, orientable, Haken 3-manifold and the fundamental group of every hyperbolic piece of the torus decomposition of M is conjugacy separable then so is the fundamental group of M. Invoking recent work of Agol and Wise, it follows that if M is a compact, orientable 3-manifold then π_1(M) is conjugacy separable.

math.GR↗
arXiv2011

Alternating quotients of free groups

We strengthen Marshall Hall's Theorem to show that free groups are locally extended residually alternating. Let F be any free group of rank at least two, let H be a finitely generated subgroup of infinite index in F and let {g_1,...,g_n} be a finite subset of F-H. Then there is a surjection f from F to a finite alternating group such that f(g_i) is not in f(H) for any i. The techniques of this paper can also provide symmetric quotients.

math.GR↗
arXiv2011

On the difficulty of presenting finitely presentable groups

We exhibit classes of groups in which the word problem is uniformly solvable but in which there is no algorithm that can compute finite presentations for finitely presentable subgroups. Direct products of hyperbolic groups, groups of integer matrices, and right-angled Coxeter groups form such classes. We discuss related classes of groups in which there does exist an algorithm to compute finite presentations for finitely presentable subgroups. We also construct a finitely presented group that has a polynomial Dehn function but in which there is no algorithm to compute the first Betti number of the finitely presentable subgroups.

math.GR↗
arXiv2011

One-ended subgroups of graphs of free groups with cyclic edge groups

Consider a one-ended word-hyperbolic group. If it is the fundamental group of a graph of free groups with cyclic edge groups then either it is the fundamental group of a surface or it contains a finitely generated one-ended subgroup of infinite index. As a corollary, the same holds for limit groups. We also obtain a characterisation of surfaces with boundary among free groups equipped with peripheral structures.

math.GR↗
arXiv2010

Polygonal words in free groups

A longstanding question of Gromov asks whether every one-ended word-hyperbolic group contains a subgroup isomorphic to the fundamental group of a closed hyperbolic surface. An infinite family of word-hyperbolic groups can be obtained by taking doubles of free groups amalgamated along words that are not proper powers. We define a set of polygonal words in a free group of finite rank, and prove that polygonality of the amalgamating word guarantees that the associated square complex virtually contains a $π_1$-injective closed surface. We provide many concrete examples of classes of polygonal words. For instance, in the case when the rank is 2, we establish polygonality of words without an isolated generator, and also of almost all simple height 1 words, including Baumslag--Solitar relator $a^p (a^q)^b$ for $pq\ne0$.

math.GR↗
arXiv2010

Some virtually special hyperbolic 3-manifold groups

Let M be a complete hyperbolic 3-manifold of finite volume that admits a decomposition into right-angled ideal polyhedra. We show that M has a deformation retraction that is a virtually special square complex, in the sense of Haglund and Wise and deduce that such manifolds are virtually fibered. We generalise a theorem of Haglund and Wise to the relatively hyperbolic setting and deduce that the fundamental group of M is LERF and that the geometrically finite subgroups of the fundamental group are virtual retracts. Examples of 3-manifolds admitting such a decomposition include augmented link complements. We classify the low-complexity augmented links and describe an infinite family with complements not commensurable to any 3-dimensional reflection orbifold.

math.GT↗