# Geometry/Topology Seminar

Thursdays (and sometimes Tuesdays) 2:00-3:00PM, in 308 Eckhart.
if you have problems viewing the abstracts click on the link Geometry/Topology Seminar Web Page

## Fall 2005

September 14 (SPECIAL DAY - Wednesday 2:00-3:00PM)
SPECIAL LOCATION - Eckhart 203
Andres Navas-Flores, IHES and Universidad de Chile
Random products of circle diffeomorphisms
 Abstract: In this talk I will consider random walks on finitely generated groups of circle homeomorphisms, proving that for "non elementary" actions (i.e. without invariant probability measure) satisfying some natural "irreducibility" condition (i.e. there is no circle homeomorphism centralizing the group), the circle (endowed with the corresponding stationnary measure) is a boundary. This corresponds to a probabilistic approach to the weak Tits alternative for groups of circle homeomorphisms proven by Margulis; we will however follow an idea introduced by Ghys. As a corollary one obtains the unicity of the stationnary measure for the action. For the case of groups of C^1 diffeomorphisms one can consider the Lyapunov exponent with respect to the Lebesgue measure: I will give the ideas of the proof of the fact that this number is negative for non elementary actions. As a corollary one obtain an improved and sharp version of a classical result for codimension one foliations by Sacksteder: every non elementary group of C^1 circle diffeomorphisms contains elements having only hyperbolic fixed points. Finally, I will briefly discuss the main problem arising in the subject, namely the identification of the Poisson boundary for groups of circle diffeomorphisms.

DOUBLE HEADER - SPECIAL DAY AND TIMES
September 16 (Friday 11:00AM-12:00PM in Eckhart 203)
Alex Eskin, University of Chicago
Quasi-isometries of solvable Lie groups and their lattices
 Abstract: I will describe a new method for studying quasi-isometries of solvable Lie groups. This method yields a proof of a long-standing conjecture concerning the group of quasi-isometries of the three dimensional, simply connected, unimodular Lie group Sol. This conjecture is known to imply that any finitely generated group quasi-isometric to Sol is (up to standard issues of finite index) a lattice in Sol. This is joint work with David Fisher and Kevin Whyte.

September 16 (Friday 2:00-3:00PM in Eckhart 203),
Bill Goldman, University of Maryland
A crash course on dynamics on representation varieties I
 Abstract: Automorphism groups of fundamental groups of surfaces (such as "mapping class groups") are mysterious groups which arise in many mathematical contexts. These groups act on spaces of representations of surface groups, preserving natural symplectic or Poisson geometries and invariant smooth measures. Depending on the representations, the mapping class group actions display diverse dynamical behavior. Complicated dynamics of the mapping class group action accompanies complicated topology of the moduli space. Two of the most basic spaces in the theory of Riemann surfaces represent extreme cases: (1) The Teichmuller space is contractible with proper action of the mapping class group. (2) The Jacobi variety is a closed manifold, with a chaotic (ergodic) action of the mapping class group. In general, the dynamics intermediates between these two extremes.

September 19 (SPECIAL DAY - Monday 2:00-3:00PM room TBA),
Bill Goldman, University of Maryland
A crash course on dynamics on representation varieties II
 Abstract: See above

October 6,
David Fisher, Indiana University
First cohomology and local rigidity
 Abstract: In 1964, Andre Weil showed that a homomorphism $\pi$ from a finitely generated group $\Gamma$ to a Lie group $G$ is {\em locally rigid} whenever $H^1(\G,\mathfrak g)=0$. Here $\pi$ is locally rigid if any nearby homomorphism is conjugate to $\pi$ by a small element of $G$, and $\mathfrak g$ is the Lie algebra of $G$. The point of my talk is describe an infinite dimensional analog of Weil's theorem: \begin{theorem}Let $\Gamma$ be a finitely presented group, $(M,g)$ a compact Riemannian manifold and $\pi:\G{\rightarrow}\Isom(M,g){\subset}\Diff^{\infty}(M)$ a homomorphism. Then if $H^1(\G,\Vect^{\infty}(M))=0$, the homomorphism $\pi$ is locally rigid as a homomorphism into $\Diff^{\infty}(M)$. \end{theorem} This work is motivated by a string of results by many authors, beginning with Zimmer, generalizing results about finite dimensional representations of lattices in Lie groups to representations taking values in diffeomorphism groups.

October 13,
Juan Souto, University of Chicago
The rank of the fundamental group of those hyperbolic 3-manifolds fibering over the circle
 Abstract: Given a pseudo-Anosov homeomorphism $\phi:S\to S$ of a closed surface $S$ then it is a Theorem of Thurston that the mapping torus $M_\phi$ of $\phi$ admits a hyperbolic metric. In this talk we explain how to determine, using methods of hyperbolic geometry, the minimal cardinality of a generating set for the fundamental group of $M_{\phi^n}$ for all sufficiently large $n$. Moreover, we prove that any two minimal generating sets are Nielsen equivalent to each other.

October 20,
Christian Bonatti, University of Bourgogne
Roots in the mapping class group
 Abstract:

October 28 (SPECIAL DAY - Friday 2:00-3:00PM in E308),
Michael Larsen, Indiana University
Representation growth for finitely generated groups
 Abstract: For $\Gamma$ a finitely generated group, we consider the growth of the sequence $a_n$ of $n$-dimensional irreducible complex representations of $\Gamma$. A measure of the growth rate of $a_n$ is the abscissa of convergence of the Dirichlet series $\sum a_n n^{-s}$. I will discuss what is known (and what is not) about the abscissa of convergence for general discrete groups, with emphasis on arithmetic groups satisfying the congruence subgroup property. This is joint work with Alex Lubotzky.

November 1 (SPECIAL DAY - Tuesday),
Maryam Mirzakhani, Princeton University
Equidistribution results in moduli spaces of quadratic differentials and train tracks
 Abstract:

November 10,
Michael Entov, Technion
"Almost homomorphisms", "almost linear" functionals and "almost measures" in symplectic topology
 Abstract: I will discuss how objects of the following sorts appear in symplectic topology and can be used to prove various rigidity results: - real-valued "almost homomorphisms" of groups of symplectomorphisms (quasi-morphisms and their generalizations); - an "almost linear" positive functional on C(M) for a symplectic manifold M, with the "almost linearity" being sensitive to the Poisson brackets; - an "almost measure" on M corresponding to such an "almost linear" positive functional by a generalization (due to Johan Aarnes) of the classical Riesz representation theorem. The talk is based on joint works with P.Biran, L.Polterovich and F.Zapolsky.

November 17,
Igor Belegradek, Georgia Institute of Technology
Aspherical manifolds and relative hyperbolization
 Abstract: I shall explain that the aspherical manifolds produced via the relative strict hyperbolization of polyhedra enjoy many group-theoretic and topological properties of open finite volume negatively pinched manifolds, including relative hyperbolicity, nonvanishing of simplicial volume, co-Hopf property, finiteness of outer automorphism group, absence of splitting over elementary subgroups, acylindricity, and diffeomorphism finiteness for manifolds with uniformly bounded simplicial volume.

November 22 (SPECIAL DAY - Tuesday),
Heegaard splittings and the Johnson filtration of the mapping class group
 Abstract: I will discuss a general method of producing calculable invariants of Heegaard splittings of 3-manifolds using homomorphic images of the mapping class group. This method will then be applied to the specific homomorphisms of the mapping class group developed by Johnson and Morita. This is a work in progress. This is a joint project with Joan Birman and Tara Brendle.

December 1,
Kevin Wortman, Yale University
Dynamics of unipotent flows on moduli space for genus 2
 Abstract: I'll speak about some work in progress with Kariane Calta in classifying measures on certain submanifolds of the moduli space of genus 2 translation surfaces that are ergodic with respect to unipotent flows.

December 2 (SPECIAL DAY - Friday),
Yair Glasner, Institute for Advanced Study
Generic actions of some hyperbolic groups
 Abstract: Question: Given a finitely generated group G. What can one say about its generic actions on a set. I will spend some time formulating this question in precise mathematical terms. I will then discuss the only case that has been previously studied - where G is a nonabelian free group. Using methods developed in a recent joint work with Souto and Storm I will give some answers to this question for new families of groups. Including locally quasiconvex hyperbolic groups and hyperbolic 3-manifold groups.

December 8,
Theron Hitchman, Rice University
Cohomology, cocycles and dynamical rigidity for lattices
 Abstract: We shall describe a differential geometric approach to the study of rigidity properties of lattices in semisimple Lie groups. We shall provide several applications. First, new cohomology vanishing results for such lattices with coefficients in certain infinite dimensional representations. Second, a geometric proof of Zimmer's measurable cocycle superrigidity theorem which includes previously unknown groups. Finally, we shall discuss new local rigidity statements actions of these groups on compact manifolds. This is joint work with David Fisher of Indiana University.

January 5,
Kamlesh Parwani, University of Houston
Fixed points for abelian actions on surfaces
 Abstract: We prove that if $F$ is a finitely generated abelian group of orientation preserving $C^1$ diffeomorphisms of $R^2$ which leaves invariant a compact set, then there is a common fixed point for all elements of $F.$ Furthermore, if $F$ is any abelian subgroup of orientation preserving $C^1$ diffeomorphisms of $S^2$, then there is a common fixed point for all elements of a subgroup of $F$ with index at most two. We will also discuss possible extensions of this result to other orientable surfaces (except the torus).

January 12,
Dan Margalit, University of Utah
TBA
 Abstract:

February 16,
Slava Grigorchuk, Texas A&M Universit
TBA
 Abstract:

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For questions, contact
• Benson Farb, farb at math.uchicago.edu
• Alex Eskin, eskin at math.uchicago.edu
• Shmuel Weinberger, shmuel at math.uchicago.edu