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- D. Cooper, M. Culler, H. Gillet, D. D. Long, P. B. Shalen
- 1994

Consider a compact 3-manifold M with boundary consisting of a single torus. The papers [CS1], [CS2] and [CGLS] discuss the variety of characters of SL2(C) representations of π1(M), and some of the ways in which the topological structure of M is reflected in the algebraic geometry of the character variety. We will describe in this paper a certain affine… (More)

- D. COOPER, D. D. LONG, A. W. REID
- 1997

A question dating back to Waldhausen [10] and discussed in various contexts by Thurston (see [9]) is the problem of the extent to which irreducible 3-manifolds with infinite fundamental group must contain surface groups. To state our results precisely, it is convenient to make the definition that a map i : S # M of a closed, orientable connected surface S… (More)

If Γ is a finite co-area Fuchsian group acting on H, then the quotient H2/Γ is a hyperbolic 2-orbifold, with underlying space an orientable surface (possibly with punctures) and a finite number of cone points. Through their close connections with number theory and the theory of automorphic forms, arithmetic Fuchsian groups form a widely studied and… (More)

We show that all closed flat n-manifolds are diffeomorphic to a cusp crosssection in a finite volume hyperbolic n+ 1-orbifold. AMS Classification 57M50; 57R99

- D Cooper, D D Long, Walter Neumann, David Gabai, Joan Birman
- 1998

It is shown that with finitely many exceptions, the fundamental group obtained by Dehn surgery on a one cusped hyperbolic 3–manifold contains the fundamental group of a closed surface. AMS Classification numbers Primary: 57M27 Secondary: 57M50, 20H10

- D. D. Long, A. W. Reid
- 2007

A well-known conjecture about closed hyperbolic 3-manifolds asserts that the first Betti number can be increased without bound by passage to finite sheeted covers. If the manifold is fibred, a strengthening of this conjecture is that the number of fibred faces (see §2.1 for the definition of a fibred face) of the unit ball of the Thurston norm can be made… (More)

This paper reviews the two variable polynomial invariant of knots de ned using representations of the fundamental group of the knot complement into SL2C: The slopes of the sides of the Newton polygon of this polynomial are boundary slopes of incompressible surfaces in the knot complement. The polynomial also contains information about which surgeries are… (More)

This sharpens [10], where it was shown that the complex length spectrum of M determines its commensurability class. Suppose M ′ is an arithmetic hyperbolic 3-manifold which is not commensurable to M . Theorem 1.1 implies QL(M) 6= QL(M ′), though by Example 2.1 below it is possible that one of QL(M ′) or QL(M) contains the other. By the length formulas… (More)

- M. Lackenby, D. D. Long, A. W. Reid
- 2006

This paper investigates properties of finite sheeted covering spaces of arithmetic hyperbolic 3-orbifolds (see §2). The main motivation is a central unresolved question in the theory of closed hyperbolic 3-manifolds; namely whether a closed hyperbolic 3-manifold is virtually Haken. Various strengthenings of this have also been widely studied. Of specific to… (More)

- DARYL COOPER, DARREN LONG
- 2012

This study of properly or strictly convex real projective manifolds introduces notions of parabolic, horosphere and cusp. Results include a Margulis lemma and in the strictly convex case a thick-thin decomposition. Finite volume cusps are shown to be projectively equivalent to cusps of hyperbolic manifolds. This is proved using a characterization of… (More)