Max Dehn Seminar
on Geometry, Topology, Dynamics, and Groups
Spring 2013
Date  Speaker  Title — click for abstract (if available) 
August 28 
Todd Fisher BYU 
Dynamical foliations for system isotopic to Anosov
diffeomophisms
We discuss the existence of foliations that are invariant
under the dynamics for systems that are isotopic to Anosov
diffeomorphisms. Specifically, we examine partially hyperbolic
diffeomorphisms with one dimensional center that are isotopic to a
hyperbolic toral automorphism and contained in a connected component.
We show in this case there is a center foliation. We will also discuss
more general cases where there is a weak form of hyperbolicity called a
dominated splitting. This is joint work with Jerome Buzzi, Rafael
Potrie, and Martin Sambarino.

September 18 
Jon Chaika
University of Utah 
Minimal and not uniquely ergodic interval exchange
transformations
This talk will introduce two well known methods for
constructing minimal and not uniquely ergodic interval exchange
transformations. Then it will discuss some recent results on minimal and
not uniquely ergodic interval exchanges focussing on diophantine
properties and connections to geometry. Joint work with J. Athreya, M.
Boshernitzan, Y. Cheung, H. Masur, P. Reynolds and M. Wolf.

October 2 
Francois
Ledrappier Notre Dame 
Regularity of the stochastic entropy under conformal changes
We consider a negatively curved closed Riemannian manifold.
The "stochastic entropy" and the "linear drift" are two numbers which
describe how the heat diffuses at infinity on the universal cover in
large times. We recall some of their properties and discuss their
regularity as functions of the metric.

October 9 
Martin
Deraux Grenoble 
Complex hyperbolic geometry of the figure eight knot
I will review Riley's construction of a complete real
hyperbolic
metric on the figure eight knot complement, and explain how it can
be modified to show that the figure eight knot complement is also
spherical CR uniformizable (i.e. it occurs as the manifold at infinity
of a well chosen discrete subgroup of automorphisms of the ball).

October 23 
Grigori
Avramidi University of Utah 
Ends of nonpositively curved manifolds
This talk will be about complete, nonpositively curved,
Riemannian manifolds that are not compact, but do have finite volume.
The topology of ends of such manifolds is well understood for the
``extreme cases'' of manifolds of pinched negative curvature and higher
rank locally symmetric spaces, but appears to be more mysterious in
general. I will describe examples of such manifolds and discuss a recent
result showing in the four dimensional case that the end is aspherical.
Joint work with T. Nguyen Phan and Y. Wu.

November 6 
Gabriela
WeitzeSchmithuesen Karlsruhe Institute of Technology 
Hurwitz translation surfaces
Finite translation surfaces are closed surfaces which carry
outside of finitely many cone points an atlas all of whose transition
maps are translations. They are obtained by gluing finitely many
polygons along parallel edges of the same length via translations. Their
natural automorphism group consists of the affine homeomorphisms of the
surface with trivial derivative, i.e. which are translations. The number
of translations of a translation surface of genus g is naturally bounded
by the RiemannHurwitz formula. We study for which g the bound is
achieved.

November 13 
Dave
Morris Lethbridge 
Strictly convex norms on amenable groups
It is obvious that the usual Euclidean norm is strictly
convex, by which we mean that, for all x and all nonzero y, either x +
y > x, or x  y > x. We will discuss the existence of such a norm
on an abstract (countable) group G. A sufficient condition is the
existence of a faithful action of G by orientationpreserving
homeomorphisms of the real line. No examples are known to show that this
is not a necessary condition, and we will combine some elementary
measure theory and dynamics with the theory of orderable groups to show
that the condition is indeed necessary if G is amenable. This is joint
work with Peter Linnell of Virginia Tech.

November 20 
Evelyn Lamb University of Utah 
The Hopf differential and harmonic maps between branched
hyperbolic structures
Some hyperbolic metrics are smooth, and some are
not. We extend Wolf's parametrization of Teichmüller space by harmonic
maps to the space of hyperbolic surfaces with cone singularities.
These surfaces give rise to
representations into the nonHitchin components of the representation
space Hom(π_1(M), PSL(2,R)).

December 4 
Sebastian
Hensel University of Chicago 
Unicorns and Beyond
In this talk, I will first present joint work with Piotr
Przytycki and
Richard Webb giving a new short proof of uniform hyperbolicity of curves
and arc graphs. Namely, I will describe unicorn paths in arc and curve
graphs
and show that they form 1slim triangles. Using this, one can
deduce that arc graphs are 7hyperbolic (and curve graphs are
17hyperbolic)
I will then overview some other results which, in a similar vein, give
quick and purely
topologicalcombinatorial proofs of curve graph results. If time
permits, I will explain
how such proofs can sometimes be adapted to work in the Out(F_n) setting
and allow
new insights there.

December 11 
Simion
Filip University of Chicago 
Teichmuller dynamics and algebraic geometry
The space of pairs (Riemann surface, holomorphic 1form)
admits an action of the group SL(2,R). This generalizes the genus one
case of the action on the tangent bundle of the modular surface. This
action is intimately related to more classical dynamical systems, such
as billiards in polygons and interval exchanges. Many questions about
these systems can be addressed using the SL(2,R) action.
Eskin and Mirzakhani recently proved a number of rigidity properties of
SL(2,R)invariant measures. In particular, these are of Lebesgueclass
on complex manifolds. In this talk I will explain the proof that these
manifolds are in fact algebraic varieties, with interesting arithmetic
properties.
I will begin by introducing some motivating examples from Teichmuller
dynamics. I will then introduce the necessary concepts from algebraic
geometry. No background in either subject will be assumed.

January 22 
Ilya
Gekhtman University of Chicago 
The stable type of the mapping class group and some
relatively hyperbolic groups.
The stable ratio set of a nonsingular action is a notion
introduced by Bowen and Nevo to prove pointwise ergodic theorems for
measure preserving actions of certain nonamenable groups.
I prove that stable ratio set of the action of a discrete subgroup of
isometries of a CAT(1) space with finite BowenMargulis measure on its
boundary with the PattersonSullivan measure has numbers other than 0, 1
and \infinity, extending techniques of Bowen from the setting of
hyperbolic groups.
I also prove the same result for the action of the mapping class group
on the sphere of projective measured foliations with the Thurston
measure, using some "statistical hyperbolicity" properties for the (non
hyperbolic) Teichmueller metric on Teichmueller space.

February 5 
Mladen Bestvina University of Utah 
On the asymptotic dimension of a curve complex
Asymptotic dimension of a metric space is a large scale
analog of the usual covering dimension. I will talk about a proof that
the curve complex of a surface has asymptotic dimension bounded by a
certain linear function of the complexity of the surface. Previously,
the known bound was exponential. This is joint work with Ken Bromberg.

February 12 
Ioannis
Konstantoulas UIUC 
Effective multiple mixing for Weyl chamber actions
A famous theorem of S. Mozes implies that a mixing action of a
semisimple group on a probability space $X$ is mixing of all orders.
Prompted by a question of D. Dolgopyat, we have studied the quantitative
behavior of multiple correlation integrals and provided explicit
exponential rates of decay for the Weyl chamber action of higher rank
groups. In this talk, we will discuss these results and their
significance, the major techniques we used (especially the use of
semidirect products, which provides a gateway for similar results in
much more general settings) and applications to equidistribution and the
search for limit laws.

February 19 
Andy Sanders UIC 
A new proof of Bowen's theorem on Hausdorff dimension of
quasicircles
A quasiFuchsian group is a discrete group of Mobius
transformations of the Riemann sphere which is isomorphic to the
fundamental group of a compact surface and acts properly on the
complement of a Jordan curve: the limit set. In 1979, Bowen proved a
remarkable rigidity theorem on the Hausdorff dimension of the limit set
of a quasiFuchsian group: it is equal to 1 if and only if the limit set
is a round circle. This theorem now has many generalizations. We will
present a new proof of Bowen's result as a byproduct of a new lower
bound on the Hausdorff dimension of the limit set of a quasiFuchsian
group. This lower bound is in terms of the differential geometric data
of an immersed, incompressible minimal surface in the quotient manifold.
If time permits, generalizations of this result to other
convexcocompact surface groups will be presented.

February 26 
Matthew
Durham UIC 
Elliptic Actions on Teichmüller Space
Kerckhoff's solution to the Nielsen realization problem showed
that the action of any finite subgroup of the mapping class group on
Teichmüller space has a fixed point. The set of fixed points is a
totally
geodesic submanifold. We study the coarse geometry of the set of points
which have bounded diameter orbits in the Teichmüller metric. We show
that
each such almostfixed point is within a uniformly bounded distance of
the
fixed point set, but that the set of almostfixed points is not
quasiconvex. In addition, the orbit of any point is shown to have a
fixed
barycenter. In this talk, I will discuss the machinery and ideas used
in
the proofs of these theorems.

March 5 
Alan Huckleberry Jacobs University and Ruhr Universitat Bochum 
Schubert intersection theory and the complex geometry of flag domains 
March 19 
Yair Glasner Ben Gurion University/University of Utah 
Invariant random subgroups of linear groups.
An invariant random subgroup (IRS for short) of a countable
group G is a
conjugation invariant probability measure on the (compact metric) space
of
all subgroups of G.
Theorem: Let G < \GL_n(F) be a countable nonamenable linear group with a simple center free Zariski closure. Then. 1. There is a topology on G such that for every IRS, almost every nontrivial subgroup of G is open. 2. G contains a dense free subgroup F in this topology. 3. For every IRS, the map taking a subgroup H < G to its intersection with F becomes an Finvariant isomorphism of probability spaces. We say that an action of a group G on a probability space is a.s.n.f if almost all point stabilizers are nontrivial. Corollary: Let G as above acts on two probability spaces X and Y. If both actions are a.s.n.f then so is the diagonal action on the product. 
April 2 
Grigori Avramidi University of Utah 
Flat tori in the homology of some locally symmetric spaces
I’ll explain why many finite covers of SL(m,Z)\SL(m,R)/SO(m)
have nontrivial homology classes generated by totally geodesic flat
(m1)tori. This is joint work with Tam Nguyen Phan.

April 9 
Matthew Stover Temple University 
Small generators for arithmetic groups
Let N = X / H be the quotient of a product of symmetric spaces
and buildings by a
cocompact Sarithmetic lattice in a reductive group G. Does H have a
generating set
whose associated geodesics on N are, say, bounded above by a polynomial
in vol(N)
depending only on dim(X)? Conjectures in number theory imply the answer
is no even
in the case where H is the unit group of a number field k (i.e., G =
GL_1 and S is
the archimedean places). I will talk about joint work with Ted Chinburg
on the case
where G is the multiplicative group of a division algebra D over a
number field k,
generalizing work of H. W. Lenstra when D = k, saying that, once S is
sufficiently
large, the answer is actually yes. Even stronger, we prove the existence
of a matrix
representation where the generators have `small' entries.

April 16 
Martin Bridson University of Oxford 10:45 LCB 121 
Volume gradient and the growth of homology in towers of
residuallyfree groups
A typical example of a residually free group is a subgroup
of a direct product of surface groups.
We are interested in how the rank of homology and the cellular volume
of classifying spaces grow as one passes
to subgroups of increasing finite index in a fixed group of geometric
interest.
I shall introduce an invariant that measures the number of cells that one needs in the kskeleta of classifying spaces for a tower of finiteindex subgroups in a group with a finite K(G,1), explaining a connection to L^2 invariants. I shall then use the structure theory of residuallyfree groups to calculate asymptotic volumeinvariants and L^2 betti numbers for such groups. This is joint work with Desi Kochloukova (Campinas). 
April 14 
Nir Lazarovich Technion 10:45 LCB 121 
Regular CAT(0) Cube Complexes
Over the past years CAT(0) cube complexes have played a major
role in
geometric group theory and have provided many examples of interesting
group actions on CAT(0) spaces. In the search for highly symmetric
CAT(0) cube complexes  just as for their 1dimensional analogues,
trees  it is natural to consider the subclass of "regular" CAT(0)
cube complexes, i.e., cube complexes with the same link at each
vertex. However, unlike regular trees, general regular CAT(0) cube
complexes are not necessarily uniquely determined by their links.
In this talk, we will discuss a necessary and sufficient condition for
uniqueness. We will then explore some examples of unique regular cube
complexes and the properties of their automorphism groups.

April 23 
Diana Davis
Northwestern University 
Cutting sequences on translation surfaces
We will investigate a dynamical system that comes from
geodesic
trajectories on flat surfaces. We will start with known results for the
square torus and the double pentagon surface, and then discuss new
results
for BouwMöller surfaces, made from many polygons.

Current seminar Archive of past talks
Max Dehn Seminar is organized by Mladen Bestvina, Ken Bromberg, Patrick Reynolds,
Jing Tao, Domingo Toledo, and Kevin Wortman.
This web page is maintained by Patrick Reynolds and Jing Tao.