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Thursday
9am, Introduction to nilpotent groups. Moon Duchin
Abstract . I’ll develop some basic tools for working with finitely-generated
nilpotent groups, with a focus on the Heisenberg group. We’ll encounter Mal’cev coordinates, Pansu’s theorems, and tools from Lie
groups and Lie algebras.
10:20am, Aspects of growth in Baumslag-Solitar groups. Eric
Freden
Abstract . We explore exponential growth rates, with specific focus on
BS(2, 4)
10:50am, Compactifications of manifolds with boundary. Shijie Gu
Abstract . In 1976, Chapman and Siebenmann presented criteria for
a Hilbert cube manifold X to admit a Z-compactification. Although
Guilbault (2001) showed that the extension of their characterization
cannot be extended to all locally compact ANR’s, the following question remains open: If M n is a finite dimensional manifold and M n × Q
is Z-compactifiable, is M n itself Z-compactifiable? In this talk, we
will provide a characterization of completable manifolds with boundary by generalizing a theorem of O’Brien (1983). More specifically, we
give necessary and sufficient conditions for M n (n > 5) to be homeomorphic to the complement of a compact subset of the boundary of a
compact manifold. As a result, we will obtain a best possible ”stabilization theorem”: M n × Q (n > 4) is Z-compactifiable if and only if
M n ×[0, 1] is Z-compactifiable. This is joint work with Craig Guilbault.
11:20am, Obstructions to Riemannian smoothings of a locally
CAT(0) manifold. Bakul Sathaye
Abstract . In this talk I will discuss obstructions to Riemannian smoothings of a locally CAT(0) manifold. I will focus on obstructions in dimension = 4 given by Davis-Januszkiewicz-Lafont and show how their
methods can be extended to construct more examples of locally CAT(0)
4-manifolds that do not support Riemannian metric with nonpositive
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sectional curvature. Further, the universal cover of such a manifold
satisfies the isolated flats condition and contains a collection of 2dimensional flats with the property that their boundaries at infinity
form a non-trivial link in the boundary of the universal cover.
1:40pm, Mapping the Braid Groups in to Lattices. Nancy Scherich
Abstract . Many of the well loved representations of the braid groups
have the property that they fix a hermitian form, i.e. the representations are unitary. With the help of Salem numbers, I will show a
process that takes in unitary representations of the braid groups and
gives representations into lattices.
2:10pm, Inscrutability. Michael Andersen
Abstract . A standard tool in algebraic topology is to pass between a
continuous map between spaces and the corresponding homomorphism
of fundamental groups using the π1 functor. It is a non-trivial question
to ask when a specific homomorphism is induced by a continuous map;
that is, what is the image of the π1 functor on homomorphisms?
2:40pm, Boundary Swapping Using Quasi-Isometries. Molly
Moran
Abstract . Bestvina defined a Z-structure on a group to formalize the
concept of a group boundary. In his original paper, he outlined a
boundary swapping condition that allows one to take the boundary
off one space and put it on a finite K(G, 1). He also suggested that
a generalized version of boundary swapping should hold for any two
groups that are quasi-isometric. We will present some of the ingredients
we need for this proof as well as some of its further generalizations. This
is joint work with Craig Guilbault.
3:30pm, Extension Theory in Large Scale Geometry. Atish Mitra
Abstract . Classical extension theory deals with extensions of maps
between topological spaces. In this talk we will discuss progress made
in the extension theory of functions in various large scale categories, and
will compare the results and techniques with that of classical extension
theory.
4:00pm, Z-Structures on Semidirect Products. Brian Pietsch
Abstract . Z-Structures were originally defined by Bestvina as an attempt to extend the notion of CAT(0)/hyperbolic group boundaries to
more generalized groups. In this talk, I will discuss a special case of
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constructing a Z-structure on a semidirect product. Specifically, when
the kernel group admits a Z-structure and the quotient group is infinite
cyclic, the semidirect product of the two will admit a Z-structure.
4:30pm, Wild homology groups. Greg Conner
Abstract . We’ll briefly discuss discuss some new results and conjectures
concerning homology and fundamental groups of non-locally simply
connected spaces in low dimensions. In particular, we will mention
Eda’s result that all interesting 1-dimensional Peano continua have
the same first homology and then we will consider the first homology
groups of compact subspaces of Euclidean 3-space and conjecture that
they either are finite rank abelian or have a free factor isomorphic to
the first homology of the Hawaiian earring.
Friday
9am, Growth series and rationality. Moon Duchin
Abstract . Jim Cannon’s classic 1984 paper gave geometric tools for
showing that hyperbolic groups had a certain ”linear recursion” property which today goes by the name of rational growth. Rationality gives
strong information about what Cannon termed the ”global combinatorial structure” of the group. Using ingredients from the last lecture,
I’ll present a proof of rational growth in the Heisenberg group.
10:20am,Monotone-light factorizations in coarse geometry. Thomas
Weighill
Abstract . A classical result of general topology states that every continuous map between compact metric spaces (and more generally, between compact Hausdorff spaces) can be factorized as a monotone map
(that is, a surjective continuous map whose fibres are connected) followed by a light map (that is, a continuous map whose fibres are totally disconnected). In coarse geometry, one is interested in studying
the large scale behaviour of spaces rather than their topology (the motivation for studying such behaviour comes mainly from index theory
and geometric group theory). In this talk we introduce large scale (or
coarse) versions of monotone and light maps, and show that these these
two classes of maps constitute a factorization system on the coarse category. For the case of maps between proper metric spaces, we exhibit
some connections between the coarse and classical notions of monotone
and light using the Higson corona. We also point out some category
theoretic connections between the classical and coarse notions. Finally,
we look at some coarse properties which are preserved by coarsely light
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maps, such as finite asymptotic dimension and exactness. This is joint
work with Jerzy Dydak.
10:50am, Distortion of surfaces in graph manifold. Hoang Nguyen
Abstract . Let S be an immersed horizontal surface in a 3-dimensional
graph manifold. We show that the fundamental group of the surface S
is quadratically distorted whenever the surface is virtually embedded
(i.e., separable) and is exponentially distorted when the surface is not
virtually embedded. This is joint work with Chris Hruska.
11:20am, Subcommensurated subgroups and finitely generated groups that are simply connected at ∞. Mike Mihalik
Abstract .
1:40pm, Examples of cyclically presented 3-manifold spines.
Kirk McDermott
Abstract . Cyclically presented groups naturally arise as the fundamental group of certain closed, orientable 3-manifolds. In this talk,
we prove a family of cyclically presented groups is a new collection of
3-manifold spines. A common approach is to take a diagram representing an essential spherical map, and then perform a classical face
pairing technique using an Euler characteristic argument. Here, we
instead work with a a spherical picture- dual to a diagram- and show,
equivalently, when the picture represents a Heegaard diagram for a
3-manifold.
2:10pm, The Transition to Asphericity. Bill Bogley
Abstract . Consider a relative two-complex (L, K) where K is an aspherical CW complex (of arbitrary dimension). The pair (L, K) is said
to be aspherical” if the second relative homotopy group of the pair
(L, K) is trivial, which is equivalent to saying that the inclusion of
K in L induces a monomorphism of fundamental groups and that the
complex L is aspherical in the traditional sense. Results obtained over
the past 25 years indicate that if one fixes the combinatorial structure
of the quotient two-complex L/K, then most pairs (L, K) tend to be
aspherical in this sense. The temptation to catalog and explore those
that are not aspherical in this relative sense is therefore inevitable and
perhaps even rational. I will survey recent work in this area, mostly
from collaborations with Forrest Parker, Kirk McDermott, and Gerald
Williams. Issues arising thus far touch on number theory, computational group theory, combinatorial group theory, and three-manifold
topology.
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2:40pm, Open problems and recent progress on understanding
ends of groups. Craig Guilbault
Abstract . We will describe and discuss several open problems involving ends of groups, most notably the Semistability Conjecture and the
Z-boundary Problem. We will also present some recent progress on
these problems, touching upon joint work with a variety of coauthors,
including: R. Ancel, R. Geoghegan, M. Mihalik, M. Moran, P. Sparks,
and C. Tirel.
3:30pm, Simplicial Inverse Sequences in Extension Theory.
Leonard Rubin
Abstract . In extension theory, in particular in dimension theory, it is
frequently useful to represent a given compact metrizable space X as
the limit of an inverse sequence of compact polyhedra. We are going to
show that, for the purposes of extension theory, it is possible to replace
such an X by a better metrizable compactum Z. This Z will come as
the limit of an inverse sequence of triangulated polyhedra with bonding
maps that are simplicial with respect to these fixed triangulations, and
that factor in a certain way. As has been proved by S. Mardešić, not
every metrizable compactum can be represented by such a “simplicial”
inverse sequence. There will be a cell-like map π : Z → X, and we
show that if K is a CW-complex with Xτ K, then Zτ K.
The simplicial inverse sequence determining Z is subject to “adjustments.” Any such adjustment determines an inverse sequence whose
limit maps onto X, and the fibers of such a map have explicit and
useful descriptions as the limits of certain subsequences. We plan to
exploit this technique as part of a unified approach involving proofs
of the resolution theorems of Edwards-Walsh (cell-like), Dranishnikov
(Z/p-acyclic), and Levin (Q-acyclic), and perhaps to obtain a new and
different one more general than the latter.
4:00pm, Translation like actions of nilpotent groups. Mark Pengitore
Abstract . We show that for nilpotent groups with the same growth
that having non-isomorphic Carnot completions is an obstruction to
the existence of translation like actions.
4:30pm, The Dunce Hat Does Not Have The Double Collapsible Property. Pete Sparks
Abstract . We show that the dunce hat cannot be decomposed into two
collapsible subcomplexes whose intersection is also collapsible. This is
related to the notion of the double n-space property of a contractible
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open n-manifold: M has the double n-space property if M can be written as the union of two euclidean n-spaces meeting in a third euclidean
n-space.
Saturday
9am, Geometry of words. Moon Duchin
Abstract . What do you know about a group from a strong understanding of its Cayley geodesics? I’ll pull questions and themes from across
geometric group theory (boundaries, equation-solving, random models,
etc) and give applications for nilpotent groups.
10:20am, Products of Thom cochains via intersection and linking forms. Greg Friedman
Abstract . If N is a smooth submanifold of a manifold M , then N
determines a cochain (which we call a Thom cochain) acting on the
singular chains transverse to N by taking intersection numbers. Cup
products (and higher products) of such cochains can then be studied in
terms of local intersection and linking information among submanifolds.
We will discuss the beginning stages of this work in progress.
10:50am, A Classification theorem for 1-Dimensional boundaries of groups with isolated flats. Matthew Haulmark
Abstract . In 2000 Kapovich and Kleiner proved that if G is a oneended hyperbolic group that does not split over a two-ended subgroup,
then the boundary of G is either a Menger curve, a Sierpinski carpet,
or a circle. Kim Ruane asked if there was a CAT(0) generalization of
Kapovich and Kleiners theorem. As boundaries of CAT(0) groups are
in general not locally connected, there is no hope of such a generalization for general CAT(0) groups. However, a version of Kapovich and
Kleiners theorem may hold for certain classes of CAT(0) groups. In
this talk I will discuss a generalization of their theorem for CAT(0)
groups with isolated flats, and provide an example of a non-hyperbolic
CAT(0) group with Menger curve boundary.