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Dynamics of the mapping class group action on the variety of PSL(2,C) characters
 math.GT/0504474 (submitted). GROUP REPRESENTATIONS 33
"... Abstract. We study the action of the mapping class group Mod(S) on the boundary ∂Q of quasifuchsian space Q. Among other results, Mod(S) is shown to be topologically transitive on the subset C ⊂ ∂Q of manifolds without a conformally compact end. We also prove that any open subset of the character va ..."
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Abstract. We study the action of the mapping class group Mod(S) on the boundary ∂Q of quasifuchsian space Q. Among other results, Mod(S) is shown to be topologically transitive on the subset C ⊂ ∂Q of manifolds without a conformally compact end. We also prove that any open subset of the character variety X(π1(S), PSL2 C) intersecting ∂Q does not admit a nonconstant Mod(S)invariant meromorphic function. This is related to a question of Goldman. 1.
A brief survey of the deformation theory of Kleinian Groups
 GEOMETRY & TOPOLOGY MONOGRAPHS VOLUME 1: THE EPSTEIN BIRTHDAY SCHRIFT PAGES 23–49
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A class of locally conformally flat 4manifolds
, 2012
"... Weconstructinfinite familiesofnonsimplyconnectedlocallyconformallyflat(LCF) 4manifolds realizing rich topological types. These manifolds have strictly negative scalar curvature and the underlying topological 4manifolds do not admit any Einstein metrics. Such 4manifolds are of particular interest ..."
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Weconstructinfinite familiesofnonsimplyconnectedlocallyconformallyflat(LCF) 4manifolds realizing rich topological types. These manifolds have strictly negative scalar curvature and the underlying topological 4manifolds do not admit any Einstein metrics. Such 4manifolds are of particular interest as examples of Bachflat but nonEinstein spaces in the nonsimply connected case. Besides that the underlying smooth manifolds are examples of spaces that admit open book decomposition in dimension 4. 1
The curious moduli spaces of unmarked Kleinian surface groups
, 2010
"... Fixing a closed hyperbolic surface S, we define a moduli space AI(S) of unmarked hyperbolic 3manifolds homotopy equivalent to S. This 3dimensional analogue of the moduli space M(S) of unmarked hyperbolic surfaces homeomorphic to S has bizarre local topology, possessing many points that are not clo ..."
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Cited by 4 (3 self)
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Fixing a closed hyperbolic surface S, we define a moduli space AI(S) of unmarked hyperbolic 3manifolds homotopy equivalent to S. This 3dimensional analogue of the moduli space M(S) of unmarked hyperbolic surfaces homeomorphic to S has bizarre local topology, possessing many points that are not closed. There is, however, a natural embedding ι: M(S) → AI(S) and compactification AI(S) such that ι extends to an embedding of the DeligneMumford compactification M(S) → AI(S).
Minimal volume Alexandrov spaces
 J. Diff. Geom
"... Abstract. Closed hyperbolic manifolds are proven to minimize volume over all Alexandrov spaces with curvature bounded below by −1 in the same bilipschitz class. As a corollary compact convex cores with totally geodesic boundary are proven to minimize volume over all hyperbolic manifolds in the same ..."
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Abstract. Closed hyperbolic manifolds are proven to minimize volume over all Alexandrov spaces with curvature bounded below by −1 in the same bilipschitz class. As a corollary compact convex cores with totally geodesic boundary are proven to minimize volume over all hyperbolic manifolds in the same bilipschitz class. Also, closed hyperbolic manifolds minimize volume over all hyperbolic cone manifolds in the same bilipschitz class with cone angles ≤ 2π. The proof uses techniques developed by BessonCourtoisGallot. In 3 dimensions, this result provides a partial solution to a conjecture in Kleinian groups concerning acylindrical manifolds. 1.
HYPERBOLIC CONVEX CORES AND SIMPLICIAL VOLUME
, 2009
"... Abstract. This paper investigates the relationship between the topology of hyperbolizable 3manifolds M with incompressible boundary and the volume of hyperbolic convex cores homotopy equivalent to M. Specifically, it proves a conjecture of Bonahon stating that the volume of a convex core is at leas ..."
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Cited by 3 (1 self)
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Abstract. This paper investigates the relationship between the topology of hyperbolizable 3manifolds M with incompressible boundary and the volume of hyperbolic convex cores homotopy equivalent to M. Specifically, it proves a conjecture of Bonahon stating that the volume of a convex core is at least half the simplicial volume of the doubled manifold DM, and this inequality is sharp. This paper proves that the inequality is in fact sharp in every pleating variety of AH(M). 1.
CONVERGENCE GROUPS, HAUSDORFF DIMENSION, AND A THEOREM OF SULLIVAN AND TUKIA
, 2006
"... We show that a discrete, quasiconformal group preserving H n has the property that its exponent of convergence and the Hausdorff dimension of its limit set detect the existence of a nonempty regular set on the sphere at infinity to H n. This generalizes a result due separately to Sullivan and Tuki ..."
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Cited by 2 (0 self)
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We show that a discrete, quasiconformal group preserving H n has the property that its exponent of convergence and the Hausdorff dimension of its limit set detect the existence of a nonempty regular set on the sphere at infinity to H n. This generalizes a result due separately to Sullivan and Tukia, in which it is further assumed that the group act isometrically on H n, i.e. is a Kleinian group. From this generalization we are able to extract geometric information about infiniteindex subgroups within certain of these groups.
RESEARCH BLOG 3/07/03
"... This week there were two talks at U. of C. on hyperbolic 3manifolds. ..."
Topology of Nonsimply connected LCF 4Manifolds
, 2008
"... We construct handlebody diagrams of families of nonsimply connected Locally Conformally Flat (LCF) 4manifolds realizing rich topological types, which are obtained from conformal compactification of the 3manifolds, that are built from the Panelled Web Groups. These manifolds have strictly negative ..."
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We construct handlebody diagrams of families of nonsimply connected Locally Conformally Flat (LCF) 4manifolds realizing rich topological types, which are obtained from conformal compactification of the 3manifolds, that are built from the Panelled Web Groups. These manifolds have strictly negative scalar curvature and the underlying topological 4manifolds do not admit any Einstein metrics. 1