Geometry/Topology Seminar
Winter 2013
Thursdays (and sometimes Tuesdays) 34pm, in
Eckhart 308

 Thursday January 10 at 3pm in Eck 308
 Kate Juschenko, Vanderbilt University
 On simple amenable groups

Abstract: We will discuss amenability of the
topological full group of a minimal Cantor system. Together
with the results of H. Matui this provides examples of
finitely generated simple amenable groups. Joint with N.
Monod.

 Thursday January 17 at 3pm in Eck 308
 Anne Thomas, Sydney University
 Infinite reduced words and the Tits boundary of a Coxeter group

Abstract: Let (W,S) be a Coxeter system with W
infinite. An infinite reduced word of W is an infinite
sequence of elements of S such that each finite subsequence
is a reduced word. We prove that the limit weak order on the
blocks of infinite reduced words of W is encoded by the
topology of the Tits boundary of the Davis complex of W. We
consider many special cases, including W word hyperbolic and
X with isolated flats. This is joint work with Thomas Lam.

 Thursday January 31 at 3pm in Eck 308
 Asaf Hadari, Yale University
 Homological shadows of attracting laminations

Abstract: Let S be a surface with
punctures, and let f \in Mod(S) be a
pseudoAnosov mapping class. Associated to f is an
attracting lamination L, which is the limit under
the forward orbit of f of any closed curve on
S. We address the following question  is there a
natural way to associate to L some natural object
in the homology of S? If so, can it be described
using some limiting process? What would such an object tell
us about f? We show that there is indeed such an
object, and that it possesses a surprising amount of
structure. For instance, if f is in the Torelli
group, then the homological lamination will be a convex
polyhedron with rational vertices. We will show pictures of
interesting phenomena that appear in these new objects, and
present applications.

 Thursday February 7 at 3pm in Eck 308
 Vaibhav Gadre, Harvard University
 Word length statistics for Teichmuller geodesics and
singularity of harmonic measures.

Abstract: Kaimanovich and Masur showed that a random
walk on the mapping class group when projected to
Teichmuller space by its action converges almost surely to
the Thurston boundary. This defines a harmonic or hitting
measure on the boundary. Restricting to finitely supported
random walks, we consider word length statistics along
Teichmuller geodesics. We show that along geodesics typical
with respect to harmonic measure word length grows linearly.
On the other hand, we also show that along geodesics typical
with respect to Lebesgue measure word length grows
superlinearly. In particular, this gives a geometric proof
of the fact that harmonic measure is singular with respect
to Lebesgue measure. This is joint work with J. Maher and G.
Tiozzo. We also discuss other contexts in which such results
hold.

 Tuesday February 12 at 3pm in Eck 308
 Daniel Cohen, LSU
 Chen ranks and resonance

Abstract: The Chen groups of a group G are the lower
central series quotients of G/G”, where G” is
the second commutator subgroup. Under certain conditions, we
relate the ranks of the Chen groups to the first resonance
variety, a jump locus for the cohomology of G. In the case
where G is the fundamental group of the complement of a
complex hyperplane arrangement, our results positively
resolve Suciu's Chen ranks conjecture. Joint work with H.
Schenck (UIUC)

 Thursday February 21 at 3pm in Eck 308
 Laurent Bartholdi, GeorgAugust University of Gottingen
 Complex dynamics and group theory 1

Abstract: The iteration of a rational map is a
straightforward, but extremely rich dynamical system. It is
often studied by analytic tools (Julia and Fatou sets), but
was also considered by Thurston as a purely topological
dynamical system; somewhat in parallel to his geometrization
results, he studied when the rational map may be recovered
from the topogical data. I will develop, in this lecture
series, a grouptheoretical language that is ideally suited
to study these questions. The fundamental idea, due to
Nekrashevych, is an equivalence between selfcoverings of a
topological space and selfsimilar group actions on rooted
trees. I will show how longstanding open problems in
complex dynamics may be answered in an almost
straightforward manner using these new tools. Part of this
work is joint with Nekrashevych and Dudko.

 Tuesday February 26 at 3pm in Eck 308
 Laurent Bartholdi, GeorgAugust University of Gottingen
 Complex dynamics and group theory 2

Abstract: The iteration of a rational map is a
straightforward, but extremely rich dynamical system. It is
often studied by analytic tools (Julia and Fatou sets), but
was also considered by Thurston as a purely topological
dynamical system; somewhat in parallel to his geometrization
results, he studied when the rational map may be recovered
from the topogical data. I will develop, in this lecture
series, a grouptheoretical language that is ideally suited
to study these questions. The fundamental idea, due to
Nekrashevych, is an equivalence between selfcoverings of a
topological space and selfsimilar group actions on rooted
trees. I will show how longstanding open problems in
complex dynamics may be answered in an almost
straightforward manner using these new tools. Part of this
work is joint with Nekrashevych and Dudko.

 Thursday February 28 at 3pm in Eck 308
 Igor Rivin, Temple University
 Random elements in (hopefully) interesting groups

Abstract: In recent times, there has been a
considerable amount of activity on the properties of
"random" elements in infinite (usually) groups. We will
sketch some of these results, and talk about the question of
how these elements might be generated.

 Thursday March 7 at 3pm in Eck 308
 Laurent Bartholdi, GeorgAugust University of Gottingen
 Complex dynamics and group theory 3

Abstract: The iteration of a rational map is a
straightforward, but extremely rich dynamical system. It is
often studied by analytic tools (Julia and Fatou sets), but
was also considered by Thurston as a purely topological
dynamical system; somewhat in parallel to his geometrization
results, he studied when the rational map may be recovered
from the topogical data. I will develop, in this lecture
series, a grouptheoretical language that is ideally suited
to study these questions. The fundamental idea, due to
Nekrashevych, is an equivalence between selfcoverings of a
topological space and selfsimilar group actions on rooted
trees. I will show how longstanding open problems in
complex dynamics may be answered in an almost
straightforward manner using these new tools. Part of this
work is joint with Nekrashevych and Dudko.

 Thursday March 14 at 3pm in Eck 308
 Dror BarNatan, University of Toronto
 Trees and Wheels and Balloons and Hoops

Abstract: Balloons are twodimensional spheres.
Hoops are one dimensional loops. Knotted Balloons and Hoops
(KBH) in 4space behave much like the first and second
fundamental groups of a topological space  hoops can be
composed like in \pi_{1}, balloons like
in \pi_{2}, and hoops "act" on balloons
as \pi_{1} acts on
\pi_{2}. We will observe that ordinary
knots and tangles in 3space map into KBH in 4space and
become amalgams of both balloons and hoops. We give an
ansatz for a tree and wheel (that is, freeLie and cyclic
word) valued invariant \zeta of KBHs in terms of
the said compositions and action and we explain its
relationship with finite type invariants. We speculate that
\zeta is a complete evaluation of the BF
topological quantum field theory in 4D, though we are not
sure what that means. We show that a certain "reduction and
repackaging" of \zeta is an "ultimate Alexander
invariant" that contains the Alexander polynomial
(multivariable, if you wish), has extremely good composition
properties, is evaluated in a topologically meaningful way,
and is leastwasteful in a computational sense. If you
believe in categorification, that's a wonderful playground.
For more information, please go to
http://www.math.toronto.edu/~drorbn/Talks/Chicago1303/

 Thursday March 21 at 3pm in Eck 308
 Melanie Matchett Wood, University of Wisconsin
 Motivic Stabilization of Configuration Spaces and Connections to Topology

Abstract: For a topological space X, the
moduli spaces of n distinct, unordered points,
Conf^{n}X, have stable Betti numbers as
n gets large, by a result of McDuff, Church, and
RandallWilliams. For an algebraic variety X, one can also
study the motives of the spaces Conf^{n}X
(roughly, the images of Conf^{n}X in the
Grothendieck group of varieties), and in recent work with
Vakil we show that these also stabilize as n gets large, in
many cases. Moreover, we give precise formulas for the
stable limit of these motives. We will discuss the
relationship between homological stabilization and motivic
stabilization, and how our results suggest questions about
the limits of the Betti numbers of
Conf^{n}X. Futher, we discuss how our
results suggest homological stabilization of many kinds of
variants on configuration spaces, as well as moduli spaces
of smooth hypersurfaces and variants.
For questions, contact