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Lawrence Reeves

Publications and source records attributed to Lawrence Reeves.

15 recordsLinked to original sources
arXiv2021

An Irrational-slope Thompson's Group

The purpose of this paper is to study the properties of the irrational-slope Thompson's group $F_τ$ introduced by Cleary in 1995. We construct presentations, both finite and infinite and we describe its combinatorial structure using binary trees. We show that its commutator group is simple. Finally, inspired by the case of Thompson's group F, we define a unique normal form for the elements of the group and study the metric properties for the elements based on this normal form. As a corollary, we see that several embeddings of $F$ in $F_τ$ are undistorted.

math.GR↗
arXiv2020

Comparing decompositions of Poincaré duality pairs

Analogues of JSJ decompositions were developed for Poincaré duality pairs in [19]. These decompositions depend only on the group. Our focus will be on describing the edge splittings of these decompositions more precisely. We use our results to compare these decompositions with two other closely related decompositions.

math.GR↗
arXiv2020

Irrational-slope versions of Thompson's groups $T$ and $V$

In this paper we consider the $T$- and $V$- versions, $T_τ$ and $V_τ$ , of the irrational slope Thompson group $F_τ$ considered in [3]. We give infinite presentations for these groups and show how they can be represented by tree-pair diagrams similar to those for $T$ and $V$. We also show that $T_τ$ and $V_τ$ have index-2 normal subgroups, unlike their original Thompson counterparts $T$ and $V$. These index-2 subgroups are shown to be simple.

math.GR↗
arXiv2019

Affine reflection subgroups of Coxeter groups

In this paper we study affine reflection subgroups in arbitrary infinite Coxeter groups of finite rank. In particular, we study the distribution of roots of Coxeter groups in the root subsystems associated with affine reflection subgroups. We give a characterization of limit roots arising from affine reflection subgroups. We also give a characterization of when a Coxeter group may possess affine reflection subgroups. We show that the intersection of the normalized isotropic cone (associated with the Tits representation of a Coxeter group) and the imaginary cone consists of limit roots closely related to affine reflection subgroups.

math.GR↗
arXiv2018

Neighbourhoods in root systems of infinite Coxeter groups

Let $W$ be a finitely generated infinite Coxeter group, with $Φ$ and $Π$ being the corresponding root system and set of simple roots respectively. It has been observed by Hohlweg et la that the projections of elements of $Φ$ onto suitably chosen hyperplanes, called \emph{normalized roots}, are contained in the convex hull of $Π$ (which is a compact set), and hence the set of all normalized roots may exhibit interesting asymptotical behaviours. In this paper we investigate the topology of the limit set of the normalized roots and demonstrate a natural system of neighbourhoods around each limit point arising from a non-affine infinite dihedral reflection subgroup of $W$.

math.GR↗
arXiv2018

Commutators in groups of piecewise projective homeomorphisms

In 2012 Monod introduced examples of groups of piecewise projective homeomorphisms which are not amenable and which do not contain free subgroups, and later Lodha and Moore introduced examples of finitely presented groups with the same property. In this article we examine the normal subgroup structure of these groups. Two important cases of our results are the groups $H$ and $G_0$. We show that the group $H$ of piecewise projective homeomorphisms of $\mathbb{R}$ has the property that $H"$ is simple and that every proper quotient of $H$ is metabelian. We establish simplicity of the commutator subgroup of the group $G_0$, which admits a presentation with $3$ generators and $9$ relations. Further we show that every proper quotient of $G_0$ is abelian. It follows that the normal subgroups of these groups are in bijective correspondence with those of the abelian (or metabelian) quotient.

math.GR↗
arXiv2017

Mapping the Davis complex into the imaginary cone

The study of the set of limit roots associated to an infinite Coxeter group was initiated by Hohlweg, Labbé and Ripoll and further developed by Dyer, Hohlweg, Péaux and Ripoll. The Davis complex associated to a finitely generated Coxeter group $W$ is a piecewise Euclidean CAT(0) space on which $W$ acts properly, cocompactly by isometries. The one skeleton of the Davis complex can be identified with the Cayley graph of $W$. In this paper we define a natural map from the Davis complex into the normalised imaginary cone of a based root system.

math.GR↗
arXiv2013

Some non-contracting automata groups

We add to the classification of groups generated by 3-state automata over a 2 letter alphabet given by Bondarenko et al., by showing that a number of the groups in the classification are non-contracting. We show that the criterion we use to prove a self-similar action is non-contracting also implies that the associated self-similarity graph introduced by Nekrashevych is non-hyperbolic.

math.GR↗
arXiv2011

On the genus of infinite groups

We associate to each finite presentation of a group G a compact CW-complex that is a 3-manifold in the complement of a point, and whose fundamental group is isomorphic to G. We use this complex to define a notion of genus for G and give examples, and also define a notion of `closed group'. A group has genus 0 if and only if it is the fundamental group of a compact orientable 3-manifold.

math.GR↗
arXiv2011

On the algorithmic construction of classifying spaces and the isomorphism problem for biautomatic groups

We show that the isomorphism problem is solvable in the class of central extensions of word-hyperbolic groups, and that the isomorphism problem for biautomatic groups reduces to that for biautomatic groups with finite centre. We describe an algorithm that, given an arbitrary finite presentation of an automatic group $Γ$, will construct explicit finite models for the skeleta of $K(Γ,1)$ and hence compute the integral homology and cohomology of $Γ$.

math.GT↗
arXiv2008

A Combination Theorem for Strong Relative Hyperbolicity

We prove a combination theorem for trees of (strongly) relatively hyperbolic spaces and finite graphs of (strongly) relatively hyperbolic groups. This gives a geometric extension of Bestvina and Feighn's Combination Theorem for hyperbolic groups and answers a question of Swarup. We also prove a converse to the main Combination Theorem.

math.GT↗
arXiv1997

Groups acting on CAT(0) cube complexes

We show that groups satisfying Kazhdan's property (T) have no unbounded actions on finite dimensional CAT(0) cube complexes, and deduce that there is a locally CAT(-1) Riemannian manifold which is not homotopy equivalent to any finite dimensional, locally CAT(0) cube complex.

math.GR↗
arXiv1995

Central Extensions of Word Hyperbolic Groups

Thurston has claimed (unpublished) that central extensions of word hyperbolic groups by finitely generated abelian groups are automatic. We show that they are in fact biautomatic. Further, we show that every 2-dimensional cohomology class on a word hyperbolic group can be represented by a bounded 2-cocycle. This lends weight to the claim of Gromov that for a word hyperbolic group, the cohomology in every dimension is bounded.

math.GR↗
arXiv1994

Regular Cocycles and Biautomatic Structures

Let $E$ be a virtually central extension of the group $G$ by a finitely generated abelian group $A$. We show that $E$ carries a biautomatic structure if and only if $G$ has a biautomatic structure $L$ for which the cohomology class of the extension is represented by an $L$-regular cocycle. Moreover, a cohomology class is $L$-regular if some multiple of it is or if its restriction to some finite index subgroup is. We also show that the entire second cohomology of a Fuchsian group is regular, so any virtually central extension is biautomatic. In particular, if the fundamental group of a Seifert fibered 3-manifold is not virtually nilpotent then it is biautomatic. ECHLPT had shown automaticity in this case and in an unpublished 1992 preprint Gersten constructed a biautomatic structure for circle bundles over hyperbolic surfaces and asked if the same could be done for these Seifert fibered 3-manifolds.

math.GR↗