98 research outputs found
Marden's Tameness Conjecture: history and applications
Marden's Tameness Conjecture predicts that every hyperbolic 3-manifold with
finitely generated fundamental group is homeomorphic to the interior of a
compact 3-manifold. It was recently established by Agol and Calegari-Gabai. We
will survey the history of work on this conjecture and discuss its many
applications.Comment: 30 pages, expository article based on a lecture given at the
conference on "Geometry, Topology and Analysis of Locally Symmetric Spaces
and Discrete Groups'' held in Beijing in July 2007. Article was published in
the proceedings of that conferenc
Topological restrictions on Anosov representations
We characterize groups admitting Anosov representations into
, projective Anosov representations into
, and Borel Anosov representations into
. More generally, we obtain bounds on the
cohomological dimension of groups admitting -Anosov representations into
and offer several characterizations of Benoist
representations
Simple length rigidity for Kleinian surface groups and applications
We prove that a Kleinian surface groups is determined, up to conjugacy in the
isometry group of , by its simple marked length spectrum. As a
first application, we show that a discrete faithful representation of the
fundamental group of a compact, acylindrical, hyperbolizable 3-manifold is
similarly determined by the translation lengths of images of elements of
represented by simple curves on the boundary of . As a second
application, we show the group of diffeomorphisms of quasifuchsian space which
preserve the renormalized intersection number is generated by the (extended)
mapping class group and complex conjugation
An improved bound for Sullivan's convex hull theorem
Sullivan showed that there exists such that if is a simply connected hyperbolic domain, then there exists a
conformally natural -quasiconformal map from to the boundary
of the convex hull of its complement which extends to the
identity on . Explicit upper and lower bounds on were
obtained by Epstein, Marden, Markovic and Bishop. We improve on these bounds,
by showing that one may choose .Comment: 24 pages, 5 figure
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