Max Dehn Seminar
on Geometry, Topology, Dynamics, and Groups
Fall 2013 — Wednesdays 3:15  4:15, Room 215
Date  Speaker  Title — click for abstract (if available) 
September 3  Yohsuke
Watanabe
University of Utah 
Local finiteness of the curve graph and its application to
Bowditch's
slice for tight geodesics
The curve graph is a locally infinite graph. In this talk, we
will show
that the curve graph could be understood as a locally finite graph via
subsurface projections.
As an application, we obtain an uniform bound of Bowditch's slices of
the
collection of all tight geodesics between any given pair of curves.

September 10 
Hyo Won Park University of Utah 
Characteristics of graph braid groups
The graph braid group over a given graph is the fundamental group
of the configuration space over the graph. We will discuss presentations
and the first homologies of graph braid groups with historical facts.

September 17 
Ben McReynolds
Purdue University 
Effective rigidity and counting
In 1992, Alan Reid proved that if two arithmetic hyperbolic
2manifolds
have the same geodesic length spectrum, the two manifolds must be
commensurable. In
2008, ChinburgHamiltonLongReid extended Reid's result to arithmetic
hyperbolic
3manifolds. In this talk, I will discuss effective versions of these
results.
Specifically, given two arithmetic hyperbolic 2 or 3manifolds of some
bounded
volume V that are not commensurable, we ensure that a length L occurs in
one but not
both. More important, the length L can be bounded above as a function of
the volume
V and is explicitly given. These results rely on effective rigidity
results for
quaternion algebras. The main tools used are algebraic and geometric
counting
results of independent interest. Time permitting, I will discuss some of
these
counting results. This work is joint with Benjamin Linowitz, Paul
Pollack, and Lola
Thompson.

September 18 3:45 in LCB 225
*Note the unusual time and usual place* 
Howard Masur
University of Chicago 
Ergodic theory of Interval exchange transformations.
An important class of dynamical systems are interval exchange
transformations. One cuts up an interval and rearranges the pieces by
translations. In the case of 2 intervals this is equivalent to a
rotation of a circle. If there are 4 or more intervals an interesting
phenonemom can happen that the the transformation is minimal but not
uniquely ergodic. This means that there is more than one invariant
measure. The set of invariant probability measures is a convex set and
the ergodic measures are the extreme points. The Birkhoff theorem says
that for an ergodic measure almost every point is generic whch means
that its orbit is uniformly distributed. A question is whether a
nonergodic measure can have such generic points. In this talk I answer
this question affirmatively with an example. This is joint work with Jon
Chaika.

September 24 
Kasra Rafi
University of Toronto 
Teichmüller space is semihyperbolic.
We provide a bicombing of Teichmüller space equipped with the
Teichmüller metric
by modifying the Teichmüller geodesic paths. As corollary, we conclude
that the Dehn functions
of the Teichmuller space are Euclidean.

October 1 
Sara Maloni
Brown University 
Polyhedra inscribed in quadrics, antide Sitter and halfpipe
geometry
In this talk we will show that a planar graph is the1skeleton of
a Euclidean polyhedron inscribed in a hyperboloid if and only if it is
the
1skeleton of a Euclidean polyhedron inscribed in a cylinder if and only
if
it is the 1skeleton of a Euclidean polyhedron inscribed in a sphere and
has a Hamiltonian cycle. This result follows from the characterisation
of
ideal polyhedra in antide Sitter and halfpipe space in terms of their
dihedral angles and induced metric on its boundary.
(This is joint work with J Danciger and JM Schlenker.)

October 8 
Tarik Aougab Yale University 
Effective geometry of curve and pants graphs
The curve and pants graphs of a surface S encode information
about hyperbolic structures on 3manifolds fibering over the circle with
fiber S. However, the way in which the geometry of these graphs
explicitly depend on S is not well understood, and this limits our
ability to use these graphs to make concrete statements about
3manifolds. In this talk we will discuss a collection of results which
shed light on this dependence. As an application, we give an effective
version of Brock's coarse volume formula, which relates the volume of a
hyperbolic 3manifold fibering over the circle with fiber S, to the
translation length of the monodromy acting on the pants graph. This is
joint work with Samuel Taylor and Richard Webb.

October 15  Fall break  
October 29 
Neil Fullarton
Rice University 
Palindromic automorphisms of free groups
The palindromic automorphism group of a free group is the group
of automorphisms that take each member of some fixed free basis to a
word that reads the same backwards as forwards. This group is an obvious
free group analogue of the hyperelliptic mapping class group of an
oriented surface. I will discuss some elementary properties of
palindromes and palindromic automorphisms, and introduce a new complex
on which the palindromic automorphism group acts. In particular, we will
discuss how the action on this complex can be used to find a generating
set for the socalled palindromic Torelli group. I will also discuss
recent joint work with Anne Thomas on generalisations of these results
to the rightangled Artin group setting.

November 5 
Rodrigo Treviño
New York University 
Flat surfaces, Bratteli diagrams and adic transformations
I will survey some recent developments in the theory of flat
surfaces of finite area and translation flows, including both compact
and (infinite genus) noncompact surfaces. In particular, I will
concentrate on a new point of view based on a joint paper with K.
Lindsey, where we develop a close connection of Bratteli diagrams and
flat surfaces. I will also state a criterion for unique ergodicity in
the spirit of Masur's criterion which holds in this very general setting
and which implies Masur's criterion in moduli spaces of (compact) flat
surfaces. No knowledge of anything will be assumed, and the talk will
not be technical and full of examples

November 12 in JTB 120 *Note the unusual location* 
Khalid BouRabee
The City College of the University of New York 
Linear groups with Borel's property
When does Borel's theorem on free subgroups of semisimple groups
generalize to other
groups? In this talk, we present a systematic study of this question and
arrive at
positive and negative answers for it. In particular, we find a full
classification
of fundamental groups of surfaces and von Dyck groups that satisfy
Borel's theorem.
We also discuss connections with a question of Breuillard, Green,
Guralnick, and Tao
concerning double word maps.

November 19 
Jing Tao University of Oklahoma 
Growth Tight Actions
Let G be a group equipped with a finite generating set S. G is
called growth tight if its exponential growth rate relative to S is
strictly greater than that of every quotient G/N with N infinite. This
notion was first introduced by Grigorchuk and de la Harpe. Examples of
groups that are growth tight include free groups relative to bases and,
more generally, hyperbolic groups relative to any generating set. In
this talk, I will provide some sufficient conditions for growth
tightness which encompass all previous known examples.

November 26  Thanksgiving eve  
December 3 
Daniel Studenmund
University of Utah 
Full residual finiteness growth of nilpotent groups
The full residual finiteness growth (FRF growth) of a group G
measures the difficulty of detecting word metric balls in G using finite
quotients of G. This is one kind of quantification of the residual
finiteness of G. We will discuss FRF growth of nilpotent groups, with
the goal of proving that FRF growth is precisely polynomial in the class
of nilpotent groups. This talk covers work joint with Khalid BouRabee.

December 10 
Kevin Wortman
University of Utah 
Word metrics and arithmetic groups
We'll talk about a nonnegative curvature result for higher rank
arithmetic groups.

January 14 
Kenneth Bromberg University of Utah 
Volumes of 3manifolds that fiber over the circle and the
translation distance of pseudoAnosovs
We will discuss a recent result of KojimaMcShane relating the
hyperbolic volume of a 3manifold that fibers over the circle to the
translation distance of the corresponding pseudoAnosov diffeomorphism.

January 21 
Ioannis Konstantoulas University of Utah 
Random lattice deformations in the Heisenberg group
We will describe ongoing work with J. Athreya on the distribution
of lattice points of a random lattice chosen uniformly with respect to
Haar measure on the space of Heisenberg lattices. The main question we
will discuss is: 'Given a set of positive measure in three dimensional
Euclidean space, what is the probability that a random Heisenberg
lattice will not hit the set?' This work extends results of Athreya and
Margulis who settled the Euclidean case and is part of a broader program
to understand statistics of lattices in a wide variety of algebraic and
Lie groups. Our methods combine the orbit structure of certain actions
of the special linear group and the continuous part of the spectral
decomposition on the space of Euclidean lattices.

January 28 
Sebastian Hensel University of Chicago 
The handlebody Torelli group
The mapping class group of a surface has various
topologically motivated subgroups. In this talk we combine two of
them: the Torelli group (of those elements acting trivially on homology)
and the handlebody group (of those elements extending to a given
handlebody).
We prove that it has an (infinite) generating set similar to the usual
Torelli group, answering a question of Joan Birman.
We also begin to develop a Johnson theory for the handlebody Torelli
group, and highlight some of the many open questions about this group.
This
is joint workinprogress with Andy Putman.

February 4 
Jenny Wilson Stanford University 
Stability phenomena for representations of classical Weyl groups
Over the past few years Church, Ellenberg, Farb, and Nagpal have
developed machinery for studying sequences of representations of the
symmetric groups, using a concept they call an FImodule. Their work
provides a theoretical framework for describing certain stability
phenomena for these sequences. I will give an overview of their theory
and describe how it generalizes to sequences of representations of the
Weyl groups in type B/C and D. I will outline some applications to
geometry and topology, including stability results for several families
of groups and spaces related to the pure braid groups.

February 11 
Domingo Toledo University of Utah 
Selfintersections of closed complex geodesics in complex
hyperbolic surfaces
A complex hyperbolic surface is a compact complex manifold X of
complex dimension 2
covered by the unit ball B^2 in C^2, that is, X is the quotient of B^2/
by a
cocompact, torsionfree lattice in PU(2,1). A closed complex geodesic
is a
compact (immersed) complex curve that is totally geodesic in the
invariant metric.
Complex hyperbolic surfaces with arithmetic fundamental group are
divided into two
classes: those that have no closed complex geodesics, and those that
have
infinitely many. The purpose of this talk is to prove that, in the
latter case,
only finitely many closed complex geodesics are embedded. A sharper
theorem will be
proved: the number of selfintersections grows proportionally to the
area. This is
in turn a consequence of an equidistribution theorem for complex
geodesics
derived from Ratner’s theorem. This is joint work with Martin Möller.

February 18 
Lei Yang
Yale University/MSRI 
Equidistribution of expanding translates of curves in homogeneous
spaces and its application to Diophantine approximation.
We consider an analytic curve $\varphi: I \rightarrow
\mathbb{M}(n\times m, \mathbb{R}) \hookrightarrow \mathrm{SL}(n+m,
\mathbb{R})$ and embed it into some homogeneous space $G/\Gamma$, and
translate it via some diagonal flow
$A=\{a(t): t > 0 \} < \mathrm{SL}(n+m,\mathbb{R})$. Under some
geometric
conditions on $\varphi$, we prove the equidistribution
of the evolution of the translated curves $a(t)\varphi(I)$ in
$G/\Gamma$,
and as a result, we prove that for almost all points on the curve, the
Dirichlet's theorem can not be improved. This is a joint work with
Nimish
Shah.

February 19 4pm LCB 219 * Note unusual time and place* 
Alex Sisto ETH Zurich 
Deviation estimates for random walks and acylindrically
hyperbolic groups
We will consider a class of groups that includes nonelementary
(relatively) hyperbolic groups, mapping class groups, many cubulated
groups and C'(1/6) small cancellation groups. Their common feature is
to admit an acylindrical action on some Gromovhyperbolic space and a
collection of quasigeodesics "compatible" with such action.
As it turns out, random walks (generated by measures with exponential
tail) on such groups tend to stay close to geodesics in the Cayley
graph. More precisely, the probability that a given point on a random
path is further away than L from a geodesic connecting the endpoints
of the path decays exponentially fast in L.
This kind of estimate has applications to the rate of escape of random
walks (local Lipschitz continuity in the measure) and its variance
(linear upper bound in the length).
Joint work with Pierre Mathieu.

February 25 
Catherine Pfaff Bielefeld University 
Dense geodesic rays in the quotient of Outer space
In 1981 Masur proved the existence of a dense Teichmueller
geodesic in moduli space. As some form of analogue, we construct dense
geodesic rays in certain subcomplexes of the Out(F_r) quotient of outer
space. This is joint work in progress with Yael AlgomKfir.

March 4 
Andrew Sale Vanderbilt University 
A geometric version of the conjugacy problem
The classic conjugacy problem of Max Dehn asks whether, for a
given group, there is an algorithm that decides whether pairs of
elements are conjugate. Related to this is the following question: given
two conjugate elements u,v, what is the shortest length element w such
that uw=wv? The conjugacy length function (CLF) formalises this
question. I will survey what is known for CLFs of groups, giving a
sketch proof for a result in semisimple Lie groups. I will also discuss
a new closely related function, the permutation conjugacy length
function (PCL), outline its potential application to studying the
computational complexity of the conjugacy problem, and describe a
result, joint with Y. Antolin, for the PCL of relatively hyperbolic
groups.

March 11 
Fanny Kassel Universite Lille 1 
Complete constantcurvature spacetimes in dimension 3
The Minkowski space R^{2,1} is the Lorentzian analogue of the
Euclidean space R^3; the antide Sitter space AdS^3 is the Lorentzian
analogue of the hyperbolic space H^3. I will survey some recent results
on the geometry and topology of their quotients by discrete groups, and
explain how the quotients of R^{2,1} by free groups (Margulis
spacetimes) are « infinitesimal analogues » of quotients of AdS^3. In
particular, we shall see that any Margulis spacetimes admits a
fundamental domain bounded by polyhedral surfaces called crooked planes.
This is joint work with J. Danciger and F. Guéritaud.

March 12 2 pm LCB 219
*Note unusual time and place* 
Andrzej Szczepanski University of Gdansk 
Outer automorphism group
of crystallographic groups
TBA

March 18  Spring Break  
March 25 
Kevin Schreve University of Wisconsin, Milwaukee 
Action dimension and L^2cohomology
The action dimension of a group G, actdim(G) is the least dimension
of a
contractible manifold which admits a proper Gaction. The action
dimension
conjecture states that the L^2cohomology of any group G vanishes above
actdim(G)/2.
I will explain the equivalence of this conjecture to the classical
Singer
conjecture. I will also explain a computation of action dimension for
rightangled
Artin groups and lattices in Euclidean buildings. This is based on
joint work with
Grigori Avramidi, Mike Davis, and Boris Okun.

April 8 
Jayadev Athreya University of Illinois UrbanaChampaign 
The ErdosSzuszTuran distribution for equivariant point
processes
We generalize a problem of ErdosSzuszTuran on diophantine
approximation to a variety of contexts, and use homogeneous dynamics to
compute an associated probability distribution on the integers. This is
joint work with Anish Ghosh.

April 15 
Thomas Schmidt Oregon State University 
Crosssections for continued fractions
We discuss settings in which interval maps are factors of
crosssections for the
geodesic flow on the unit tangent bundle of a hyperbolic surface. That
the regular
continued fraction, or Gauss, map is of this type was shown by
AdlerFlatto and by
Series, in the 1980s. With P. Arnoux, we recently affirmatively
answered a
question of LuzziMarmi by showing that the infinite family of socalled
Nakada
alphacontinued fractions is each of this type. In this talk, we will
sketch the
background, address the main points of that result, and report on
further work.

April 20 RTG Seminar in LCB 222 at 3pm  Marc Burger
University of Utah 
On volumes of representations
In many instances one can define the notion of volume of a
representation of the fundamental group of a closed manifold M into a
simple (noncompact) Lie group G. This is so for instance if M is a
surface and the symmetric space associated to G is hermitian, that is
carries an invariant 2form, or if M is a 3manifold and G is a
complex group, equivalently the associated symmetric space carries an
invariant 3form. When M is not compact the definition of volume of a
representation presents interesting difficulties; in this talk we will
show how bounded cohomology can be used to define an invariant
generalizing the volume of a representation and we will see how this
invariant is connected with the deformation theory of such
representations. This is joint work with Michelle Bucher and
Alessandra Iozzi.

April 22 
Jon Fickenscher Princeton University 
A Bound of Boshernitzan
In 1985, Boshernitzan showed that a minimal symbolic dynamical
system with a linear complexity bound must have a finite number of
probability invariant ergodic measures. We will discuss methods to
sharpen this bound in general and provide cases in which the bound may
already be reduced. This is ongoing work with Michael Damron.

April 29 
Ric Wade University of Utah 
Splittings of free groups via systems of surfaces
We describe a correspondence between splittings (BassSerre
decompositions) of a free group of rank n over finitely generated
subgroups and systems of surfaces in a doubled handlebody. One can use
this to describe a family of hyperbolic complexes on which Out(F_n)
acts. This is joint work with Camille Horbez.
