Max Dehn Seminar
on Geometry, Topology, Dynamics, and Groups
Fall 2015 — Tuesdays 2:00  3:00, Room LCB 222
Date  Speaker  Title — click for abstract (if available) 
August 26

Donald Robertson
University of Utah 
Finite products sets and minimally almost periodic groups
Bergelson and Furstenberg have shown that every positive density
subset of an amenable, minimally almost periodic group contains a
sequence
and all its finite products without repetition. In this talk I will
present joint work with V. Bergelson, C. Christopherson and P.
ZorinKranich that strengthens Bergelson and Furstenberg's result and
describes a trichotomy of amenable groups relating representation theory
with combinatorial properties of positive density subsets.

September 9

Camille Horbez
University of Utah 
Subgroups of the automorphism group of a free product
The main goal of the talk will be to present the following
version of the
Tits alternative for the outer automorphism group of a free product. Let
G_i be freely indecomposable countable groups, and let
G=G_1*...*G_k*F_N.
If all G_i and Out(G_i) satisfy the Tits alternative, then so does
Out(G).
Along the proof, I will also discuss classification results for
subgroups
of Out(F_N). This is partly joint work with Vincent Guirardel.

September 16

Howard Masur
University of Chicago 
Quasiisometric rigidity of Teichmuller space
In this talk I will consider the Teichmuller space of a surface
of genus at least 2 equipped with the Teichmuller metric and its
quasiisometry group. Studying the quasiisometry group of a space
has quite a rich history and after introducing the concepts I will
review some of this history. Then I will discuss the qi rigidity of
Teichmuller space. This is joint work with Alex Eskin and Kasra Rafi.

3:10 September 18 JWB 335
*Note the unusual time and place* 
Steven Sam
University of Wisconsin 
Commutative algebra of functor categories
Commutative algebra can be studied in the general context of a
symmetric monoidal
abelian category. I will explain some joint work with Andrew Putman and
Andrew
Snowden in this direction on the category of functors from a category
made out of
sets or vector spaces to the category of modules over a ring. In some
cases, there
are strong analogies: a version of the Hilbert basis theorem,
rationality of
Hilbert series, etc. Time permitting, I will indicate some applications
to twisted
homological stability for families of groups such as finite linear
groups.

3:10 pm October 2 LCB 219
*Note the unusual time and place* 
Kathryn Mann
University of California at Berkeley 
Automatic continuity for homeomorphism groups
To what extent does the algebraic structure of a topological
group determine its topology? Many (but not all) examples of real Lie
groups G have a unique Lie group structure, meaning that every abstract
isomorphism G > G is necessarily continuous. In this talk, I'll
discuss a recent much stronger result for groups of homeomorphisms of
manifolds: every homomorphism from Homeo(M) to any other separable
topological group is necessarily continuous.
This is part of a broader program to show that the topology and
algebraic structure of the group of homeomorphisms (or diffeomorphisms)
of a manifold M are intimately linked, and also deeply connected to the
topology of M itself. Time permitting, we'll discuss applications in
geometric topology, groups acting on manifolds, and connections with a
new program to study the quasiisometry type of homeomorphism groups.

October 7

Mladen Bestvina
University of Utah 
Isometric rigidity of the complex of free factors
The fundamental theorem of projective geometry states that
any function that sends points to points and lines to lines and
preserves incidence (essentially) comes from a projective
transformation. We prove the same theorem in the context of free
groups. This is joint work with Martin Bridson.

October 14

No Seminar: Fall break  
October 21

Ric Wade
University of British Columbia 
Subspace arrangements and BNS invariants.
We introduce a natural chain complex associated to a collection
of subspaces of a vector spaces, and discuss the associated homology. We
will give some background on BieriNeumannStrebel invariants of groups,
and show how the BNS invariant of a group leads to a nice subspace
arrangement, whose associated homology is (yet) another invariant of the
group. This can give a useful way of distinguishing between finitely
presented groups  we will give some examples involving rightangled
Artin groups.

October 28

Mark Hughes
Brigham Young University 
Braided cobordisms and the braid rank of a knot
We describe a new technique to recast geometric problems
involving the 4ball and ribbon genera of a link in terms of algebraic
properties of a braid representative. These techniques make use of
braided cobordisms, and require the study of certain shortest word
problems in the braid group described by Rudolph. This leads to a new
algebraic invariant of the knot. We will present a bound on the solution
of this shortest word problem coming from the Dehornoy leftinvariant
ordering on the braid group.

November 4

Morgan Cesa
University of Utah 
Dehn functions of arithmetic groups
The Dehn function of a finitely presented group measures the
number of relations required to reduce a word which represents the
identity to the trivial word. A wellknown conjecture states that higher
rank Sarithmetic groups have quadratic Dehn functions. An Sarithmetic
group sits as a lattice in a product of simple Lie groups.
BestvinaEskinWortman showed that if the G is an Sarithmetic group,
and
the ambient Lie group containing G has at least 3 factors, then the Dehn
function of G is bounded by a polynomial. In this talk, I will use a
specific example to illustrate the proof that one can extend the result
of
BestvinaEskinWortman to cover certain cases where the ambient Lie
group
has 2 factors.

November 11 JTB 320
*Note unusual place* 
Grace Work
University of Illinois UrbanaChampaign 
Transversals to horocycle flow on the moduli space of translation
surfaces
Computing the distribution of the gaps between slopes of saddle
connections is a question that was studied first by Athreya and Cheung
in the case
of the torus, motivated by the connection with Farey fractions, and then
in the case
of the golden L by Athreya, Chaika, and Lelievre. Their strategy
involved
translating the question of gaps between slopes of saddle connections
into return
times under horocycle flow on the space of translation surfaces to a
specific
transversal. We show how to use this strategy to explicitly compute the
distribution
in the case of the octagon, the first case where the Veech group had
multiple cusps,
how to generalize the construction of the transversal to the general
Veech case
(both joint work with Caglar Uyanik), and how to parametrize the
transversal in the
case of a generic surface in $\mathcal{H}(2)$.

3:15 November 23 LCB 225 *Note unusual day* 
Julien Paupert
Arizona State University 
Real reflections, commutators and crossratios in complex
hyperbolic space
A real reflection in complex hyperbolic space is an
antiholomorphic
involution (the prototype of such a map is complex conjugation in each
coordinate).
We provide a concrete criterion to determine whether or not a
2generator subgroup of SU(2,1) is generated by real reflections. As an
application we show that the Picard modular groups SU(2,1,O_d) are
generated by real reflections when d=1,2,3,7,11. This is joint work with
Pierre Will.

November 25

No Seminar: Thanksgiving  
December 9

Daniel Thompson
The Ohio State University 
Entropy for generalized betatransformations
Generalized betatransformations are the class of piecewise
continuous interval maps given by taking the betatransformation x↦ beta
x (mod1), where beta > 1, and replacing some of the branches with
branches of constant negative slope. We would like to describe the set
of beta for which these maps can admit a Markov partition. We know that
beta (which is the exponential of the entropy of the map) must be an
algebraic number. Our main result is that the Galois conjugates of such
beta have modulus less than 2, and the modulus is bounded away from 2
apart from the exceptional case of conjugates lying on the real line.
This extends an analysis of Solomyak for the case of
betatransformations, who obtained a sharp bound of the golden mean in
that setting.
I will also describe a connection with some of the results of Thurston's
fascinating final paper, where the Galois conjugates of entropies of
postcritically finite unimodal maps are shown to describe a beautiful
fractal. These numbers are included in the setting that we analyze.

January 19

Thibaut Dumont
University of Utah 
Cocycle growth for the Steinberg representation
The study of the growth of cocycles of compactly generated
locally compact groups leads to interesting cohomology theories, e.g.
bounded cohomology and polynomially bounded cohomology. In 2006 Monod
designated Klingler's cocycle as a good candidate for a `quasification'
procedure of Shalom and himself, and asked to determine its growth. This
natural cocycle sits in the only nontrivial class of SL_n(F) for the
Steinberg representation, where F is a finite extension of the field of
padic numbers. In this seminar, I will describe the Steinberg
representation and Klingler's construction of the cocycle in terms of
the
BruhatTits building of SL_n(F). Then I will discuss how to compute the
norm of the cocycle and present the case of SL_2(F). In rank two,
I was lead to study the relative position of three
points in the BruhatTits building of SL_3(F).

3:15 February 3 LCB 219 RTG *Note the unusual place and time* 
Burt Totaro
UCLA 
Group cohomology and algebraic cycles
The classifying space BG of a compact Lie group G (such as a
finite
group) can be viewed as a direct limit of complex algebraic varieties.
As a
result, it makes sense to consider the "Chow ring"
of algebraic cycles on BG. This ring maps to the cohomology ring of BG,
but they are usually not the same. The Chow ring seems to have a close
relation to the cobordism of BG. We survey what is known.

February 16

Bruce Kleiner
New York University 
Rigidity of product structure and biLipschitz homeomorphisms
of Carnot groups
There are many contexts where one has a mapping between
products, and one would like to know if the product structure must be
respected by the map. In geometric group theory, for instance, one
would like to know if a quasiisometry f : H_1 x H_2 ——> G_1 x G_2
between products of groups must asymptotically look like a product of
quasiisometries. The main focus of the lecture will be on the case
when the groups are nilpotent, in which case the question is tied with
a similar question for biLipschitz homeomorphisms between Carnot
groups. It turns out that currents are a key tool for addressing this
question.
This is joint with Stefan Muller and Xiangdong Xie.

February 23

Jason Behrstock
Lehman College 
Asymptotic dimension of mapping class groups
The goal of this talk will be to describe our recent result
proving that the asymptotic dimension of the mapping class group of a
closed surface is at most quadratic in the genus (building on and
strengthening a prior result of BestvinaBromberg giving an
exponential estimate). We obtain this result as a special case of a
result about the asymptotic dimension of a general class of spaces,
which we call hierarchically hyperbolic; this class includes
hyperbolic spaces, mapping class groups, Teichmueller spaces endowed
with either the Teichmuller or the WeilPetersson metric, fundamental
groups of nongeometric 3manifolds, RAAGs, etc. We will discuss the
general framework and a sketch of how this machinery provides new
tools for studying special subclasses, such as mapping class groups.
The results discussed are joint work with Mark Hagen and Alessandro
Sisto.

March 8

Caglar Uyanik
UIUC 
Relative currents and atoroidal subgroups of Out(F)
The study of outer automorphism group of a free group is closely
related to the study of mapping class groups of hyperbolic surfaces. In
this talk, I will draw analogies between these two groups, and deduce
several structural results about subgroups of Out(F) by proving several
northsouth dynamics type results. Part of this talk is based on joint
work with Matt Clay.

March 15

No Seminar: Spring break  
March 21 Distinguished Lecture Series LCB 215 3:154:14
*Note unusual time and place*

University of Michigan 
Windows, Cores and Skinning Maps
Thurston’s skinning map encodes crucial information about the
relationship between the asymptotic geometry and the internal geometry
of an infinite volume hyperbolic 3manifold. In the acylindrical case,
Thurston’s Bounded Image Theorem asserts that the skinning map has
bounded image. This result played a role in Thurston’s original proof of
his Geometrization Theorem.
In this talk, we will discuss a generalization of Thurston’s result to
the setting of deformation spaces of hyperbolic 3manifolds with freely
decomposable fundamental group. The key new tool is a Uniform Core
Theorem for sequences of homeomorphic hyperbolic 3manifolds. We will
also discuss potential applications. This is joint work with Ken
Bromberg, Jeff Brock and Yair Minsky.

March 22

Paul Apisa
University of Chicago 
Rational billiards, hyperelliptic curves, and dynamics on moduli
space
Every holomorphic oneform on a Riemann surface corresponds to a
collection of planar polygons with sides identified by translations – a
translation surface. The action of GL(2,R) on the plane induces an
action on planar polygons and hence on the moduli space of holomorphic
oneforms. Work of Eskin, Mirzakhani, Mohammadi, and Filip establishes
that GL(2,R) orbit closures are complex subvarieties. We will classify
GL(2,R) orbit closures of dimension greater than three in hyperelliptic
components of strata and verify a conjecture of Mirzakhani  that higher
rank orbit closures arise from loci of branched covers – in these
components. As corollaries, we will derive finiteness results for
geometrically primitive Teichmuller curves in hyperelliptic components
and discuss applications to the illumination and finite blocking
problems in rational billiards.

March 29

Michael Wolf
Rice University 
SL(2,C) Higgs bundles with smooth spectral data
We show that for every nonelementary representation of a surface group
into SL(2,C) there is a Riemann surface structure such that the Higgs
bundle associated to the representation lies outside the discriminant
locus of the Hitchin fibration. Along the way in the argument, we
encounter a number of constructions in the geometry of surfaces: complex
projective structures, pleated surfaces, harmonic maps to Rtrees and
the Thurston compactification. Much of the talk will be devoted to
explaining the statement and as many of the constructions as time
permits. (Joint with Richard Wentworth.)

March 30 3:154:15 LCB 323
*Note unusual time and place*

Andy Huang
Rice University 
Handle crushing harmonic maps between surfaces
We construct polynomial growth harmonic maps from oncepunctured Riemann
surfaces of
any finite genus to any evensided, regular, ideal polygon in the
hyperbolic plane.
We also establish their uniqueness within a class of exponentially
decaying
variations. Previously, harmonic maps from the complex plane have been
parameterized
by holomorphic quadratic differentials. Our harmonic maps, mapping a
genus g>1
punctured surface to a ksided polygon, correspond to meromorphic
quadratic
differentials with one pole of order (k+2) at the puncture and (4g +k2)
zeros
(counting multiplicity) on the Riemann surface domain.

April 5

Katrin Trent
Universitat Muenster 
Free groups and multiply sharply 2 and 3transitive groups
Finite sharply 2 and 3transitive groups were classified in
the 1930 by Zassenhaus and were shown to essentially arise
as $AGL_1(K)$ or $PGL_2(K)$ for some field $K$. It remained an open
question whether a similar result holds for infinite
groups. I will answer this question.

April 12

Amir Mohammadi
University of Texas 
Geodesic planes in hyperbolic 3manifolds
In this talk we discuss the possible closures of geodesic planes
in a hyperbolic 3manifold M.
When M has finite volume Shah and Ratner (independently) showed that a
very strong rigidity
phenomenon holds, and in particular such closures are always properly
immersed submanifolds of M with finite area.
We show that a similar rigidity phenomenon holds for a class of infinite
volume manifolds.
The proof uses elements from hyperbolic geometry and Margulis' approach
in the proof of the Oppenheim conjecture.
This is a joint work with C. McMullen and H. Oh.

April 19

Wolfram Hojka
Brigham Young University 
Dynamics, wild topology, and abelian groups
We calculate the first homology group of wild topological spaces, like
the Hawaiian earring and the harmonic archipelago. To that end, we make
use of a result on the combinatorial dynamics of permuting intervals in
a linear order, and relate a topological peculiarity to classical
notions from abelian group theory.

April 26

David Aulicino
University of Chicago 
Weak Mixing for Translation Surfaces with Intermediate Orbit
Closures
Work of Avila and Forni established weak mixing for the generic
straight line flow on generic translation surfaces, and the work of
Avila and Delecroix determined when weak mixing occurs for the straight
line flow on a Veech surface. Following the work of Eskin, Mirzakhani,
Mohammadi, which proved that the orbit closure of every translation
surface has a very nice structure, one can ask how the orbit closure
affects the weak mixing of the straight line flow. In this talk all of
the necessary background on translation surfaces and weak mixing will be
presented followed by the answer to this question. This is a joint work
in progress with Artur Avila and Vincent Delecroix.

May 3

Anish Ghosh
Tata Institute for Fundamental Research 
Discrepancy estimates for rational points on varieties
I will discuss some problems relating to the statistics of the
distribution of rational points on homogeneous varieties of semisimple
algebraic groups. Joint work with Alexander Gorodnik and Amos Nevo.
