Alan W. Reid
Current Organizations / Institutes
The organizations/institutes from the scholar's latest article
Research Subjects
Alan W. Reid has a total of 35 co-authors. Here are the top co-authors by number of shared publications:
- Darren D. Long
Department of Mathematics, University of California, Santa Barbara, California 93106
11 shared publications - Christopher J. Leininger
Department of Mathematics University of Illinois at Urbana‐Champaign 1409 W. Green St Urbana IL 61801 USA
4 shared publications - Martin R. Bridson
University of Oxford, UK
4 shared publications - Gaven J. Martin
Marsden Fund. Institute for Advanced Study New Zealand Massey University Auckland New Zealand
2 shared publications - Alexander Lubotzky
The Weizmann Institute of Science, Rehovot, Israel
1 shared publication - C. Maclachlan
Department of Mathematics, University of Aberdeen, Aberdeen, Scotland
1 shared publication
Latest Publications
- 2 August 2017
- Published by Wiley in Bulletin of the London Mathematical Society
- Vol. 49 (5) , 831-841
Profinite rigidity and surface bundles over the circle
- 2 August 2017
- Published by Wiley in Bulletin of the London Mathematical Society
- Vol. 49 (5) , 831-841
Profinite rigidity and surface bundles over the circle
- 20 March 2017
- Published by Taylor & Francis in Experimental Mathematics
- Vol. 27 (4) , 395-413
Principal Congruence Links: Class Number Greater than 1
In a previous article, we started an enumeration of the finitely many levels for which a principal congruence manifold can be a link complement in S3. In this article we give a complete enumeration of all the principal congruence link complements in S3, together with their levels in the case when the class number of is greater than 1.
- 20 March 2017
- Published by Taylor & Francis in Experimental Mathematics
- Vol. 27 (4) , 395-413
Principal Congruence Links: Class Number Greater than 1
In a previous article, we started an enumeration of the finitely many levels for which a principal congruence manifold can be a link complement in S3. In this article we give a complete enumeration of all the principal congruence link complements in S3, together with their levels in the case when the class number of is greater than 1.
- Open Access
Constructing 1-cusped isospectral non-isometric hyperbolic 3-manifolds
We construct infinitely many examples of pairs of isospectral but non-isometric $1$-cusped hyperbolic $3$-manifolds. These examples have infinite discrete spectrum and the same Eisenstein series. Our constructions are based on an application of Sunada's method in the cusped setting, and so in addition our pairs are finite covers of the same degree of a 1-cusped hyperbolic 3-orbifold (indeed manifold) and also have the same complex length-spectra. Finally we prove that any finite volume hyperbolic 3-manifold isospectral to the figure-eight knot complement is homeomorphic to the figure-eight knot complement.
Open AccessConstructing 1-cusped isospectral non-isometric hyperbolic 3-manifolds
We construct infinitely many examples of pairs of isospectral but non-isometric $1$-cusped hyperbolic $3$-manifolds. These examples have infinite discrete spectrum and the same Eisenstein series. Our constructions are based on an application of Sunada's method in the cusped setting, and so in addition our pairs are finite covers of the same degree of a 1-cusped hyperbolic 3-orbifold (indeed manifold) and also have the same complex length-spectra. Finally we prove that any finite volume hyperbolic 3-manifold isospectral to the figure-eight knot complement is homeomorphic to the figure-eight knot complement.
- Open Access
Profinite rigidity, fibering, and the figure-eight knot
We establish results concerning the profinite completions of 3-manifold groups. In particular, we prove that the complement of the figure-eight knot $S^3-K$ is distinguished from all other compact 3-manifolds by the set of finite quotients of its fundamental group. In addition, we show that if $M$ is a compact 3-manifold with $b_1(M)=1$, and $\pi_1(M)$ has the same finite quotients as a free-by-cyclic group $F_r\rtimes\mathbb{Z}$, then $M$ has non-empty boundary, fibres over the circle with compact fibre, and $\pi_1(M)\cong F_r\rtimes_\psi\mathbb{Z}$ for some $\psi\in{\rm{Out}}(F_r)$.Open AccessProfinite rigidity, fibering, and the figure-eight knot
We establish results concerning the profinite completions of 3-manifold groups. In particular, we prove that the complement of the figure-eight knot $S^3-K$ is distinguished from all other compact 3-manifolds by the set of finite quotients of its fundamental group. In addition, we show that if $M$ is a compact 3-manifold with $b_1(M)=1$, and $\pi_1(M)$ has the same finite quotients as a free-by-cyclic group $F_r\rtimes\mathbb{Z}$, then $M$ has non-empty boundary, fibres over the circle with compact fibre, and $\pi_1(M)\cong F_r\rtimes_\psi\mathbb{Z}$ for some $\psi\in{\rm{Out}}(F_r)$. - Open Access
Frattini and related subgroups of Mapping Class Groups
Let $\Gamma_{g,b}$ denote the orientation-preserving Mapping Class Group of a closed orientable surface of genus $g$ with $b$ punctures. For a group $G$ let $\Phi_f(G)$ denote the intersection of all maximal subgroups of finite index in $G$. Motivated by a question of Ivanov as to whether $\Phi_f(G)$ is nilpotent when $G$ is a finitely generated subgroup of $\Gamma_{g,b}$, in this paper we compute $\Phi_f(G)$ for certain subgroups of $\Gamma_{g,b}$. In particular, we answer Ivanov's question in the affirmative for these subgroups of $\Gamma_{g,b}$.Open AccessFrattini and related subgroups of Mapping Class Groups
Let $\Gamma_{g,b}$ denote the orientation-preserving Mapping Class Group of a closed orientable surface of genus $g$ with $b$ punctures. For a group $G$ let $\Phi_f(G)$ denote the intersection of all maximal subgroups of finite index in $G$. Motivated by a question of Ivanov as to whether $\Phi_f(G)$ is nilpotent when $G$ is a finitely generated subgroup of $\Gamma_{g,b}$, in this paper we compute $\Phi_f(G)$ for certain subgroups of $\Gamma_{g,b}$. In particular, we answer Ivanov's question in the affirmative for these subgroups of $\Gamma_{g,b}$. - Open Access
Determining Fuchsian groups by their finite quotients
Let $\C(\Gamma)$ be the set of isomorphism classes of the finite groups that are homomorphic images of $\Gamma$. We investigate the extent to which $\C(\Gamma)$ determines $\Gamma$ when $\Gamma$ is a group of geometric interest. If $\Gamma_1$ is a lattice in ${\rm{PSL}}(2,\R)$ and $\Gamma_2$ is a lattice in any connected Lie group, then $\C(\Gamma_1) = \C(\Gamma_2)$ implies that $\Gamma_1$ is isomorphic to $\Gamma_2$. If $F$ is a free group and $\Gamma$ is a right-angled Artin group or a residually free group (with one extra condition), then $\C(F)=\C(\Gamma)$ implies that $F\cong\Gamma$. If $\Gamma_1<{\rm{PSL}}(2,\Bbb C)$ and $\Gamma_2< G$ are non-uniform arithmetic lattices, where $G$ is a semi-simple Lie group with trivial centre and no compact factors, then $\C(\Gamma_1)= \C(\Gamma_2)$ implies that $G \cong {\rm{PSL}}(2,\Bbb C)$ and that $\Gamma_2$ belongs to one of finitely many commensurability classes. These results are proved using the theory of profinite groups; we do not exhibit explicit finite quotients that distinguish among the groups in question. But in the special case of two non-isomorphic triangle groups, we give an explicit description of finite quotients that distinguish between them.Open AccessDetermining Fuchsian groups by their finite quotients
Let $\C(\Gamma)$ be the set of isomorphism classes of the finite groups that are homomorphic images of $\Gamma$. We investigate the extent to which $\C(\Gamma)$ determines $\Gamma$ when $\Gamma$ is a group of geometric interest. If $\Gamma_1$ is a lattice in ${\rm{PSL}}(2,\R)$ and $\Gamma_2$ is a lattice in any connected Lie group, then $\C(\Gamma_1) = \C(\Gamma_2)$ implies that $\Gamma_1$ is isomorphic to $\Gamma_2$. If $F$ is a free group and $\Gamma$ is a right-angled Artin group or a residually free group (with one extra condition), then $\C(F)=\C(\Gamma)$ implies that $F\cong\Gamma$. If $\Gamma_1<{\rm{PSL}}(2,\Bbb C)$ and $\Gamma_2< G$ are non-uniform arithmetic lattices, where $G$ is a semi-simple Lie group with trivial centre and no compact factors, then $\C(\Gamma_1)= \C(\Gamma_2)$ implies that $G \cong {\rm{PSL}}(2,\Bbb C)$ and that $\Gamma_2$ belongs to one of finitely many commensurability classes. These results are proved using the theory of profinite groups; we do not exhibit explicit finite quotients that distinguish among the groups in question. But in the special case of two non-isomorphic triangle groups, we give an explicit description of finite quotients that distinguish between them. - Open Access
Nilpotent completions of groups, Grothendieck pairs, and four problems of Baumslag
Two groups are said to have the same nilpotent genus if they have the same nilpotent quotients. We answer four questions of Baumslag concerning nilpotent completions. (i) There exists a pair of finitely generated, residually torsion-free-nilpotent groups of the same nilpotent genus such that one is finitely presented and the other is not. (ii) There exists a pair of finitely presented, residually torsion-free-nilpotent groups of the same nilpotent genus such that one has a solvable conjugacy problem and the other does not. (iii) There exists a pair of finitely generated, residually torsion-free-nilpotent groups of the same nilpotent genus such that one has finitely generated second homology $H_2(-,\Z)$ and the other does not. (iv) A non-trivial normal subgroup of infinite index in a finitely generated parafree group cannot be finitely generated. In proving this last result, we establish that the first $L^2$ betti number of a finitely generated parafree group of rank $r$ is $r-1$. It follows that the reduced $C^*$-algebra of the group is simple if $r\ge 2$, and that a version of the Freiheitssatz holds for parafree groups.Open AccessNilpotent completions of groups, Grothendieck pairs, and four problems of Baumslag
Two groups are said to have the same nilpotent genus if they have the same nilpotent quotients. We answer four questions of Baumslag concerning nilpotent completions. (i) There exists a pair of finitely generated, residually torsion-free-nilpotent groups of the same nilpotent genus such that one is finitely presented and the other is not. (ii) There exists a pair of finitely presented, residually torsion-free-nilpotent groups of the same nilpotent genus such that one has a solvable conjugacy problem and the other does not. (iii) There exists a pair of finitely generated, residually torsion-free-nilpotent groups of the same nilpotent genus such that one has finitely generated second homology $H_2(-,\Z)$ and the other does not. (iv) A non-trivial normal subgroup of infinite index in a finitely generated parafree group cannot be finitely generated. In proving this last result, we establish that the first $L^2$ betti number of a finitely generated parafree group of rank $r$ is $r-1$. It follows that the reduced $C^*$-algebra of the group is simple if $r\ge 2$, and that a version of the Freiheitssatz holds for parafree groups. - 9 July 2013
- Published by Cambridge University Press (CUP) in Mathematical Proceedings of the Cambridge Philosophical Society
- Vol. 155 (3) , 459-463
Collisions at infinity in hyperbolic manifolds
For a complete, finite volume real hyperbolic n-manifold M, we investigate the map between homology of the cusps of M and the homology of M. Our main result provides a proof of a result required in a recent paper of Frigerio, Lafont and Sisto.
- 9 July 2013
- Published by Cambridge University Press (CUP) in Mathematical Proceedings of the Cambridge Philosophical Society
- Vol. 155 (3) , 459-463
Collisions at infinity in hyperbolic manifolds
For a complete, finite volume real hyperbolic n-manifold M, we investigate the map between homology of the cusps of M and the homology of M. Our main result provides a proof of a result required in a recent paper of Frigerio, Lafont and Sisto.
- 24 April 2013
- Published by Mathematical Sciences Publishers in Algebraic & Geometric Topology
- Vol. 13 (3) , 1273-1298
Restricting the topology of 1–cusped arithmetic 3–manifolds
- 24 April 2013
- Published by Mathematical Sciences Publishers in Algebraic & Geometric Topology
- Vol. 13 (3) , 1273-1298
Restricting the topology of 1–cusped arithmetic 3–manifolds
- 24 February 2011
- Published by Mathematical Sciences Publishers in Algebraic & Geometric Topology
- Vol. 11 (1) , 605-624
Commensurators of finitely generated nonfree 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.Comment: 15 pages, 1 figur
- 24 February 2011
- Published by Mathematical Sciences Publishers in Algebraic & Geometric Topology
- Vol. 11 (1) , 605-624
Commensurators of finitely generated nonfree 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.Comment: 15 pages, 1 figur