Simon's conjecture for 2-bridge knots (2009)
Boileau, Michel, Boyer, Steve, Reid, Alan W., Wang, Shicheng
It is conjectured that for each knot $K$ in $S^3$, the fundamental group of its complement surjects onto only finitely many distinct knot groups. Applying character variety theory we obtain an...
The genus spectrum of a hyperbolic 3-manifold (2009)
McReynolds, D. B., Reid, Alan W.
In this article we study the spectrum of totally geodesic surfaces of a finite volume hyperbolic 3-manifold. We show that for arithmetic hyperbolic 3-manifolds that contain a totally geodesic...
A combination theorem for Veech subgroups of the mapping class group, Geom (2008)
Christopher J. Leininger, Alan W. Reid
In this paper we prove a combination theorem for Veech subgroups of the mapping class group analogous to the first Klein-Maskit combination theorem for Kleinian groups in which two Fuchsian subgroups...
Commensurability of Fuchsian groups and their axes (2008)
James W. Anderson, Alan W. Reid
The purpose of this note is to prove the following Theorem. This answers a question raised during the problem session at the Special Session on Geometric Function Theory of the 898th meeting of the...
Covering Spaces of Arithmetic 3-Orbifolds (2008)
Lackenby, Marc, Long, Darren D., Reid, Alan W.
Let G be an arithmetic Kleinian group, and let O be the associated hyperbolic 3-orbifold or 3-manifold. In this paper, we prove that, in many cases, G is large, which means that some finite index...
A note on Trace-Fields of Kleinian Groups (2006)
Let Γ Kleinian group of finite covolume. In the past there seems to have been some confusion as to whether the trace-field of Γ is an invariant of the commensurability class of Γ. In fact, when Γ...
Commensurability classes of 2-bridge knot complements (2006)
Reid, Alan W., Walsh, Genevieve S.
We show that a hyperbolic 2-bridge knot complement is the unique knot complement in its commensurability class. We also discuss constructions of commensurable hyperbolic knot complements and put...
Length and eigenvalue equivalence (2006)
Leininger, Christopher J., McReynolds, D. B., Neumann, Walter D., Reid, Alan W.
Two Riemannian manifolds are called eigenvalue equivalent when their sets of eigenvalues of the Laplace-Beltrami operator are equal (ignoring multiplicities). They are (primitive) length equivalent...
Covering spaces of arithmetic 3-orbifolds (2006)
Lackenby, Marc, Long, Darren D., Reid, Alan W.
Let G be an arithmetic Kleinian group, and let O be the associated hyperbolic 3-orbifold or 3-manifold. In this paper, we prove that, in many cases, G is large, which means that some finite index...
A combination theorem for Veech subgroups of the mapping class group (2004)
Leininger, Christopher J., Reid, Alan W.
In this paper we prove a combination theorem for Veech subgroups of the mapping class group analogous to the first Klein-Maskit combination theorem for Kleinian groups in which two Fuchsian subgroups...
The co-rank conjecture for 3-manifold groups (2002)
Leininger, Christopher J., Reid, Alan W.
In this paper we construct explicit examples of both closed and non-compact finite volume hyperbolic manifolds which provide counterexamples to the conjecture that the co-rank of a 3-manifold group...
The co-rank conjecture for 3-manifold groups (2002)
Christopher J. Leininger, Alan W. Reid
Abstract In this paper we construct explicit examples of both closed and non-compact finite volume hyperbolic manifolds which provide counterexamples to the conjecture that the co-rank of a...
The arithmetic hyperbolic 3-manifold of smallest volume (2001)
Chinburg, Ted, Friedman, Eduardo, Jones, Kerry N., Reid, Alan W.
Reid, Alan W., Wang, Shicheng, Zhou, Qing
We address a conjecture that $\pi_1$-surjective maps between closed aspherical 3-manifolds having the same rank on $\pi_1$ must be of non-zero degree. The conjecture is proved for Seifert manifolds,...
The Bianchi groups are subgroup separable on geometrically finite subgroups (1998)
Agol, Ian, Long, Darren D., Reid, Alan W.
We show that for certain arithmetic groups, geometrically finite subgroups are the intersection of finite index subgroups containing them. Examples are the Bianchi groups and the Seifert-Weber...
Commensurability of Fuchsian groups and their axes (1997)
Anderson, James W., Reid, Alan W.
The purpose of this note is to prove the following Theorem. This answers a question raised during the problem session at the Special Session on Geometric Function Theory of the 898th meeting of the...