Geometry/Topology Seminar
Spring 2009
Thursdays (and sometimes Tuesdays) 2-3pm, in
Eckhart 308
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- Thursday April 2 at 2pm in E308
- Bruno Klingler, Institut de Mathématiques de Jussieu
- On local rigidity for complex hyperbolic lattices and non-Abelian Hodge theory
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Abstract: One central problem in the study of
lattices in simple real Lie groups is certainly the
structure of complex hyperbolic lattices and their finite
dimensional representations. Except a few examples the only
known representations of such a lattice are those obtained
by restricting representations of the ambient group
SU(n,1) (n>1). Thus let
L be a (cocompact) lattice and r: SU(n,1)
-> G be a morphism into a real simple Lie group.
Can we obtain new representations of L by
deforming the restriction of r to L?
In the 90's Goldman and Millson proved this is not possible
for the standard embedding of SU(n,1) in
SU(n+1,1). I will explain how non-Abelian Hodge
theory provides a general answer to this problem.
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- Thursday April 9 at 2pm in E308
- Jason Manning, University of Buffalo
- Geometric Dehn filling of high dimensional hyperbolic manifolds
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Abstract: We describe geometric methods for
understanding the group theoretic Dehn fillings of a high
dimensional cusped hyperbolic manifold. Such fillings are
shown to act geometrically on either CAT(-1)
spaces or CAT(0) spaces with isolated flats
depending on the type of filling performed. The shape of the
boundary is described and group theoretic information is
inferred. This is joint work with Koji Fujiwara.
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- Thursday April 16 at 2pm in E308
- Saul Schleimer, University of Warwick
- Two results on the disk complex
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Abstract: We will answer a question of Souto and Luo
and verify a conjecture of Abrams and Masur concerning the
disk complex. The solutions require topological techniques,
developed with Masur, applied to the geometry of the disk
complex.
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- Thursday April 23 at 2pm in E308
- Ivonne Ortiz, Miami University, Ohio
- Lower Algebraic K-Theory of Hyperbolic Reflections Groups
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Abstract: This is joint work with J.-F Lafont and
Bruce Magurn. I will present the lower algebraic K-theory of
the integral group ring of (all) the
3-dimensional hyperbolic reflection groups. These
groups are Coxeter groups arising as lattices
O+(3,1), with fundamental domain a
finite volume geodesic polyhedron P in
H3 (possibly with some ideal
vertices).
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- Thursday April 30 at 2pm in E308
- Alan Reid, University of Texas
- LERF and The Lubotzky Sarnak Conjecture
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Abstract: We prove that every closed hyperbolic
3-manifold has a family of (possibly infinite sheeted)
coverings with the property that the Cheeger constants in
the family tend to zero. This is used to show that, if in
addition the fundamental group of the manifold is LERF, then
it satisfies the Lubotzky-Sarnak conjecture. This is joint
with M. Lackenby and D. D. Long.
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- Tuesday May 5 at 2pm in E308
- Jean-François Lafont, Ohio State University
- Obstructions to Riemannian smoothing for certain CAT(0)-manifolds
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Abstract: I will discuss examples of closed
manifolds supporting locally CAT(0)-metrics, but which do
not support any Riemannian metric of non-positive sectional
curvature. This is work in progress with Mike Davis,
building on previous joint work with Tom Farrell.
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- Thursday May 14 at 2pm in E308
- Anne Thomas, Cornell University
- Lattices for Platonic complexes and Fuchsian buildings
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Abstract: A polygonal complex X is
Platonic if its automorphism group acts transitively on the
flags (vertex, edge, face) in X. Compact examples
include the boundaries of Platonic solids. Noncompact
examples X with nonpositive curvature were
classified by Swiatkowski, who also determined when
G=Aut(X) is nondiscrete. For example, there is a
unique X with the link of each vertex the
Petersen graph, and in this case G is
nondiscrete. A Fuchsian building is a two-dimensional
hyperbolic building. We study lattices in automorphism
groups of Platonic complexes and Fuchsian buildings. Using
similar methods for both cases, we construct uniform and
nonuniform lattices in G=Aut(X). We also show
that for some cases the set of covolumes of lattices in
G is nondiscrete, and that G admits
lattices which are not finitely generated. In fact our
results apply to the larger class of Davis complexes, which
includes examples in dimension > 2.
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- Thursday May 21 at 2pm in E308
- Tullia Dymarz, Yale University
- Bilipschitz equivalence is not equivalent to quasi-isometric equivalence
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Abstract: We give an example of two finitely
generated quasi-isometric groups that are not bilipschitz
equivalent. The proof involves combining quasi-isometric
rigidity theorems, analysis of bilipschitz maps of the
n-adics and uniformly finite homology.
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- Thursday May 28 at 2pm in E308
- Alexander Lytchak, University of Bonn
- Regularity of spaces with curvature bounded above
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Abstract: My talk is based on a joint work with
Koichi Nagano. Spaces with curvature bounded above that are
geodesically complete and locally compact have many
similarities with Alexandrov spaces with a lower curvature
bound (for example tangent cones, first variation formula
and so on). However, deeper topological results about
Alexandrov spaces such as local conicality and topological
stability proven by Pereleman are just not true for spaces
with an upper curvature bound. For instance, there are two
dimensional examples not admitting a triangulation. In some
natural situation we are able to prove topological
regularity. In the talk I will concentrate on the geometric
regularity of spaces with an upper curvature bound and
extendible geodesics. As the main application I will discuss
a characterization of topological manifolds amongst such
spaces.
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- Thursday June 4 at 2pm in E308
- Eugene Gutkin, Copernicus University and IMPAN
- Security for closed riemannian surfaces
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Abstract: A smooth riemannian manifold is secure if
all of the geodesic segments between any pair of points in
it can be blocked by a finite number of point obstacles. The
conjecture is that a compact riemannian manifold is secure
iff it is flat. In a joint work with Victor Bangert we
establish this for surfaces of genus greater than zero.
For questions, contact