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A. W. Reid

Publications and source records attributed to A. W. Reid.

13 recordsLinked to original sources
arXiv2025

Absolute profinite rigidity, direct products, and finite presentability

We prove that there exist finitely presented, residually finite groups that are profinitely rigid in the class of all finitely presented groups but not in the class of all finitely generated groups. These groups are of the form $Γ\times Γ$ where $Γ$ is a profinitely rigid 3-manifold group; we describe a family of such groups with the property that if $P$ is a finitely generated, residually finite group with $\widehat{P}\cong\widehat{Γ\timesΓ}$ then there is an embedding $P\hookrightarrowΓ\timesΓ$ that induces the profinite isomorphism; in each case there are infinitely many non-isomorphic possibilities for $P$.

math.GR↗
arXiv2022

Most Hitchin representations are strongly dense

We prove that generic Hitchin representations are strongly dense: every pair of non commuting elements in their image generate a Zariski-dense subgroup of SL_n(R). The proof uses a theorem of Rapinchuk, Benyash-Krivetz and Chernousov, to show that the set of Hitchin representations is Zariski-dense in the variety of representations of a surface group in SL_n(R).

math.GR↗
arXiv2020

Absolute profinite rigidity and hyperbolic geometry

We construct arithmetic Kleinian groups that are profinitely rigid in the absolute sense: each is distinguished from all other finitely generated, residually finite groups by its set of finite quotients. The Bianchi group $\mathrm{PSL}(2,\mathbb{Z}[ω])$ with $ω^2+ω+1=0$ is rigid in this sense. Other examples include the non-uniform lattice of minimal co-volume in $\mathrm{PSL}(2,\mathbb{C})$ and the fundamental group of the Weeks manifold (the closed hyperbolic $3$-manifold of minimal volume).

math.GT↗
arXiv2020

On the profinite rigidity of triangle groups

We prove that certain Fuchsian triangle groups are profinitely rigid in the absolute sense, i.e. each is distinguished from all other finitely generated, residually finite groups by its set of finite quotients. We also develop a method based on character varieties that can be used to distinguish between the profinite completions of certain groups.

math.GR↗
arXiv2018

Determining hyperbolic 3-manifolds by their surfaces

In this article, we prove that the commensurability class of a closed, orientable, hyperbolic 3-manifold is determined by the surface subgroups of its fundamental group. Moreover, we prove that there can be only finitely many closed, orientable, hyperbolic 3-manifolds that have the same set of surfaces.

math.GT↗
arXiv2015

Frattini and related subgroups of Mapping Class Groups

Let $Γ_{g,b}$ denote the orientation-preserving Mapping Class Group of a closed orientable surface of genus $g$ with $b$ punctures. For a group $G$ let $Φ_f(G)$ denote the intersection of all maximal subgroups of finite index in $G$. Motivated by a question of Ivanov as to whether $Φ_f(G)$ is nilpotent when $G$ is a finitely generated subgroup of $Γ_{g,b}$, in this paper we compute $Φ_f(G)$ for certain subgroups of $Γ_{g,b}$. In particular, we answer Ivanov's question in the affirmative for these subgroups of $Γ_{g,b}$.

math.GT↗
arXiv2009

Commensurators of non-free finitely generated Kleinian groups

Suppose G is a non-free finitely generated Kleinian group without parabolics which is not a lattice and let C(G) denote the commensurator in PSL(2,C). We prove that if the limit set of G is not a round circle, then C(G) is discrete. Furthermore, G has finite index in C(G) unless G is a fiber group in which case C(G) is a lattice.

math.GT↗
arXiv2008

LERF and the Lubotzky-Sarnak conjecture

We prove that every closed hyperbolic 3-manifold has a family of (possibly infinite sheeted) coverings with the property that the Cheeger constants in the family tend to zero. This is used to show that, if in addition the fundamental group of the manifold is LERF, then it satisfies the Lubotzky-Sarnak conjecture.

math.GT↗
arXiv2007

Heegaard genus and Property 'tau' for hyperbolic 3-manifolds

We show that any finitely generated non-elementary Kleinian group has a co-final family of finite index normal subgroups with respect to which it has Property $τ$. As a consequence, any closed hyperbolic 3-manifold has a co-final family of finite index normal subgroups for which the infimal Heegaard gradient is positive.

math.GT↗
arXiv2004

Generalized Dedekind sums

Classical Dedekind sums are connected to the modular group through the construction of a (Dedekind) symbol on the cusp set of the modular group. In this paper we study generalizations of Dedekind symbols and sums that can be associated to certain Fuchsian groups uniformizing 1-punctured tori.

math.GT↗
arXiv1995

Arithmeticity, Discreteness and Volume

We give an arithmetic criterion which is sufficient to imply the discreteness of various two-generator subgroups of $PSL(2,{\bold C})$. We then examine certain two-generator groups which arise as extremals in various geometric problems in the theory of Kleinian groups, in particular those encountered in efforts to determine the smallest co-volume, the Margulis constant and the minimal distance between elliptic axes. We establish the discreteness and arithmeticity of a number of these extremal groups, the associated minimal volume arithmetic group in the commensurability class and we study whether or not the axis of a generator is simple.

math.DG↗