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Eric Chesebro (Rice and Montana),
Right angled ideal polyhedra and virtually special 3-manifolds
We use the work of Haglund and Wise to prove that the fundamental group of
any finite-volume 3-manifold with a right angled ideal cell decomposition
has a finite index subgroup which is a word-quasiconvex subgroup of a
right-angled Coxeter group. As a result, all such 3-manifolds are virtually
fibered and LERF. This gives new examples of 3-manifolds which are
virtually fibered and not fibered. Joint with
Jason Deblois and Henry Wilton
Sean Cleary (CCNY, CUNY), Weak almost convexity and tame combing
conditions
Cannon introduced the notion of almost convexity for a Cayley graph of
a group, developing effective algorithms for understanding the
geometry of the Cayley graph. Though wide classes of groups are almost
convex, there are a number of weaker notions of almost convexity
satisfied by yet more groups. Mihalik and Tschantz introduced the
notion of tame 1-combings for Cayley complexes, and there are
connections between the very strongest notions of tame combings and
the strongest notions of almost convexity. We answer questions about
possible further connections between these two notions by showing that
there are groups which satisfy some of the strongest tame combing
conditions which do not satisfy even the weakest non-trivial almost
convexity conditions. Examples include some Baumslag-Solitar groups
and Thompson's group F. This is joint work with Susan Hermiller,
Melanie Stein and Jennifer Taback.
Daryl Cooper (UCSB), Manifolds, Real Projective
Geometry and the Hyper-Reals
Abstract: We will discuss the Thurston boundary of Teichmuller space
from the viewpoint of non-standard mathematics and then discuss
an analog for real projective structures on manifolds and orbifolds.
David Futer (Temple), Cusp volume of fibered 3-manifolds
Abstract: Consider a 3-manifold M that fibers over the circle, with
fiber a punctured surface F. I will discuss some recent results that
estimate the volume of a maximal cusp of M (in the hyperbolic metric)
directly from combinatorial properties of the monodromy map phi: F ->
F. These results can be applied to estimate the volume certain knot
complements: for example, the complements of closed 3-braids.
Craig Guilbault (Wisconsin), A solution to de Groot's absolute cone conjecture
A compactum X is an absolute suspension if for any pair of points
x,y in X, the space X is homeomorphic to a suspension with x and y
corresponding to the suspension points. Similarly, X is an absolute
cone if, for each point x in X, the space X is homeomorphic to a cone
with x corresponding to the cone point.
At the 1971 Prague Symposium, J. de Groot made the following two conjectures:
Conjecture.
Every n-dimensional absolute suspension is homeomorphic to the n-sphere.
Conjecture.
Every n-dimensional absolute cone is homeomorphic to an n-cell.
In 1974, Szymanski proved the 'absolute suspension conjecture' in the
affirmative for n=1,2 or 3. The remaining cases remain open. In 2005,
Nadler presented solutions to the 'absolute cone conjecture' in dimensions 1
and 2.
In this talk, we will discuss issues related to the openness of the suspension
conjecture for large n. More significantly, we will present our recently
obtained solution to the absolute cone conjecture. In particular, the
conjecture is true for n less than or equal to 4 and false for n
greater than or equal to 5.
Jesse Johnson (Yale), Common stabilizations for Heegaard splittings
Abstract: A Heegaard surface is a surface embedded in a 3-manifold so
that it cuts the 3-manifold into simple pieces called handlebodies.
Given a Heegaard surface, there is a construction called stabilization
that creates a new Heegaard surface of higher genus. It has long been
known that any two Heegaard surfaces for the same manifold can be
stabilized some number of times to produce isotopic surfaces, but it is
not well understood how many stabilizations are, in general, necessary.
I will describe how intuition from hyperbolic geometry and harmonic
surfaces led to a topological approach to understanding common
stabilizations.
Effie Kalfagianni (Michigan State), Polynomial invariants for links in
3-manifolds.
Abstract: I will discuss recent progress in obtaining polynomial
invariants for links in rational homology 3-spheres
and applications of these invariants.
Aaron Magid (University of Michigan), The Topology of Deformation
Spaces of Hyperbolic 3-Manifolds
Abstract: For any closed surface S, the deformation space AH(S) is the
space of all marked hyperbolic 3-manifolds homotopy equivalent to
S. After reviewing some of the classical results that describe
topology of the interior of AH(S), we will show that there are certain
points on the boundary where AH(S) is not locally connected. This is a
generalization of Ken Bromberg's result that the space of Kleinian
punctured torus groups is not locally connected.
Tara Mecham (Oklahoma), Hyperbolic groups which fiber in infinitely
many ways.
Abstract. We construct examples of CAT(0), free-by-cyclic, hyperbolic
groups which fiber in infinitely many ways over Z. The construction
involves adding a specialized square 2-cell to a non-positively
curved, squared 2-complex defined by labeled oriented graphs. The
fundamental groups of the resulting complexes are CAT(0), hyperbolic,
free-by-cyclic and can be mapped onto Z in infinitely many ways.
Mike Mihalik (Vanderbilt), JSJ Decompositions of Coxeter Groups
The idea of "JSJ-decompositions" for 3-manifolds began with work of
Waldhausen and was developed later through work of Jaco, Shalen and
Johansen. It was shown that there is a finite collection of 2-sided,
incompressible tori that separate a given closed irreducible
3-manifold into pieces with strong topological structure.
Sela
introduced the idea of JSJ-decompositions for groups, an idea that has
flourished in a variety of directions. The general idea is to consider
a certain class X of groups and splittings of groups in X by groups in
another class Y. E.g. Rips and Sela considered splittings of finitely
presented groups by infinite cyclic groups. For an arbitrary group G
in X the goal is to produce a unique graph of groups decomposition T
of G with edge groups in Y so that T reveals all splittings of G by
groups in Y. More specifically, if V is a vertex group of T then
either there is no Y-group that splits both G and V, or V has a
special "surface group-like" structure. It is standard to call vertex
groups of the second type "orbifold groups".
For a finitely generated Coxeter system (W,S) we produce a (reduced)
JSJ-decomposition T for splittings of W over virtually abelian
subgroups. We show T is unique with each vertex and edge group
generated by a subset of S (and so T is "visual"). The construction of
T is algorithmic. If V, a subset of S, generates an orbifold vertex
group of T then V is the disjoint union of K and M, where < M > is
virtually abelian, < K > is virtually a closed surface group or
virtually free and < V > is the direct product of < M > and < K >.
Fujiwara-Papasoglu have a more general JSJ decomposition result for
finitely presented groups over slender splittings. We compare these
decompositions.
Panos Papasoglu (Athens),
Title: Topology of the boundary and splittings
Abstract: Stallings ends theorem has been generalized to splittings
over 2 ended groups for hyperbolic groups by Bowditch using
the boundary of the group. Such splittings correspond to
local cut points of the boundary. One might hope to generalize this
further and give a topological characterization of splittings of hyperbolic
groups over any subgroups- at least generically.
We present a negative and a positive result related
to this question. (joint work with T. Delzant)
Kim Ruane (Tufts), Automorphisms of right-angled Coxeter groups.
In this talk, I will try to convince you that the automorphism group of
a right-angled Coxeter group is an interesting group to study. We begin
with some standard examples to show how these groups are, in some ways,
easier to study than other automorphism groups (i.e. Aut(F_n) for
example) yet even the some of the most elementary questions concerning
these groups are not yet answered. I will also discuss ongoing joint
work with Adam Piggott concerning these groups.
Mark Sapir (Vanderbilt), Almost all 1-related groups with at least
three generators are virtually residually (finite p-groups)
I am going to talk about joint work with A. Borisov and I. Kozakova
proving the result from the title.
Juan Souto (University of Michigan, Ann Arbor), Geometric limits of
knot complements.
Abstract: I will discuss joint results with Jessica Purcell and Richard
Kent showing that while surprisingly many hyperbolic 3-manifolds
arise as geometric limits of knot complements, there are also non-obvious
geometric and hyperbolic obstructions to be such a limit.
Genevieve Walsh (Tufts), Commensurability of knot complements
Two three-manifolds are commensurable if they admit homeomorphic
finite-sheeted covers. Here we investigate the commensurability classes of
hyperbolic 3-manifolds. In particular we show that if K is a hyperbolic
knot without hidden symmetries, there are at most three knot complements in
the commensurability class of S3- K. There are only two knots
in the tables which admit hidden symmetries, so this is conjecturally the
generic case. We also give a characterization of cyclically commensurable
knots which are periodic. This is joint work with Michel Boileau and Steve
Boyer.
Beginning graduate workshops. There will be three lectures aimed at
beginning graduate students, on the topics of geometric group theory,
3-manifolds, and hyperbolic geometry, respectively.
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