abstract

Geometric and Probabilistic Methods in Group Theory and Dynamical Systems

Texas A&M, Department of Mathematics, College Station, TX, November 3-6, 2005

ABSTRACTS

ABÉRT, Miklós, University of Chicago
Random actions on trees
Rudder 301, Thursday, November 3, 4:40-5:10

We analyze how the probabilistic method can be used to understand groups acting on regular (rooted and unrooted) trees. In particular, we explore the structure of a random group acting on such a tree. The talk is partly about joint works with Balint Virag and Yair Glasner.

ATHREYA, Jayadev, University of Chicago
Quantitative recurrence and large deviations for Teichmüller geodesic flow
Rudder 308, Friday, November 4, 5:25-5:45

We prove quantitative recurrence and large deviations results for Teichmüller geodesic flow on the moduli space of Q_g of holomorphic unit-area quadratic differentials on a compact genus g > 1 surface.

BARNHILL, Angela Kubena, The Ohio State University
Actions and related special decompositions of Coxeter groups
Rudder 308, Saturday, November 5, 5:45-6:05

Mihalik and Tschantz showed that given any action of a Coxeter group on a simplicial tree, there is an associated decomposition of the Coxeter group as a graph of special subgroups. We extend this result to higher dimensions. In particular, we consider actions of Coxeter groups on CAT(0) spaces and construct related decompositions as complexes of special subgroups.

BAUMSLAG, Gilbert, City University of New York
TBA
Rudder 301, Thursday, November 3, 9:00-9:40

Abstract not available

BEHRSTOCK, Jason, University of Utah
Thick metric spaces, relative hyperbolicity, and quasi-isometric rigidity
Rudder 308, Saturday, November 5, 5:20-5:40

In this talk we will introduce a new quasi-isometry invariant of metric spaces which we call thick. We show that any thick metric space is not (strongly) relatively hyperbolic with respect to any non-trivial collection of subsets. Further, we show that the property of being (strongly) relatively hyperbolic with thick peripheral subgroups is a quasi-isometry invariant. The class of thick groups includes many important examples such as mapping class groups of all surfaces (except those few that are virtually free), the outer automorphism group of the free group on at least 3 generators, SL_n(Z) with n>2, and others. We shall also discuss some ways in which thick groups behave rigidly under quasi-isometries. The results in this talk are joint work with C. Druţu and L. Mosher.

BESTVINA, Mladen, University of Utah
On the QI rigidity of right angled Artin groups
Rudder 301, Thursday, November 3, 2:50-3:30

Let S be a finite graph (1-dimensional simplicial complex). Form the group G(S) by introducing a generator for every vertex of S and the commuting relation whenever the vertices are joined by an edge. For example, when S_k is a set of k>1 vertices and no edges then G(S_k) is a free group F_k of rank k. When C_k is the cone on S_k, then G(C_k) = F_kxZ. Also, G(S_k*S_m)=F_kxF_m. These examples already display a lack of rigidity, since the QI type of F_k, F_kxZ, F_kxF_m does not depend on k,m>1.

Theorem. Let S, S' be two atomic graphs (no 3- or 4-cycles, no separating vertices nor edges nor stars of vertices). If G(S) and G(S') are quasi-isometric, then S and S' are isomorphic graphs.

This is joint work with Bruce Kleiner and Michah Sageev.

BONDARENKO, Ievgen, Texas A&M University
Uniform measure on limit spaces
Rudder 302, Saturday, November 5, 5:45-6:05

We study geometrical and measure-theoretical properties of limit spaces of self-similar contracting groups. In particular we prove that the measure of the tile is integral, which is a generalization of a known result for self-affine tiles. (Joint work with R. Kravchenko).

BOWEN, Lewis, Indiana University
A Generalization of the Prime Geodesic Theorem to Counting Conjugacy Classes of Free Subgroups
Rudder 302, Friday, November 4, 4:00-4:20

The prime geodesic theorem gives an asymptotic formula for the number of conjugacy classes of elements of pi₁(M) of geometric translation length at most x (where M is hyperbolic) as x tends to infinity. We generalize this to conjugacy classes of free subgroups. There are many open questions.

BRIDSON, Martin, Imperial College London
Tubular groups, snowflake words and isoperimetric spectra
Rudder 301, Friday, November 4, 5:50-6:30

In this talk I shall describe some of the remarkably rich geometry enjoyed by the fundamental groups of compact 2-complexes built from tori and annuli. In particular, I shall explain the results of joint work with Brady, Forrester and Shankar that on the snowflake groups in the above class, an understanding of which sheds light on the structure of the set of possible exponents of regular and higher-order Dehn functions.

CIOBANU, Laura, University of Auckland, New Zealand
Examples of retracts in a free group that are not the fixed subgroup of any group of automorphisms
Rudder 501, Thursday, November 3, 5:45-6:05

Let F be a free group of rank at least three. We show that some of the retracts of F studied by Martino-Ventura are not equal to the fixed subgroup of any group of automorphisms of F. This shows that, in F, there exist subgroups that are equal to the fixed subgroup of some set of endomorphisms but are not equal to the fixed subgroup of any set of automorphisms. Moreover, we determine the Galois monoids of these retracts, where by the Galois monoid of a subgroup H of F we mean the monoid of all endomorphisms of F that fix H.

DANI, Pallavi, University of Oklahoma
The asymptotic density of finite-order elements in infinite groups
Rudder 302, Thursday, November 3, 5:20-5:40

Consider a finitely generated infinite group G. Let P be a property that elements of G might have. A natural question is, how likely is it that a specific element of G has the property P? This is measured by the asymptotic density of the set of elements with property P. Fix a word metric on G. Then the asymptotic density is defined as the limit, as r tends to infinity, of the proportion of elements in the ball of radius r which have the property P. We compute the asymptotic density of finite-order elements in two main classes of groups, virtually nilpotent groups and word hyperbolic groups, for which the answers are very different.

DIACONIS, Persi, Stanford University
Gelfand pairs
Rudder 301, Saturday, November 5, 4:30-5:10

In case after case, when symmetries allow a tractable Fourier analysis, a Gelfand pair and spherical functions lie hidden. A recent attempt to extend Kirilov's "orbit method" to upper triangular matrices is used as a running example by there are many other special cases (joint work with I. M. Issacs).

ESKIN, Alex, University of Chicago
Quasi-isometric rigidity of SOL
Rudder 301, Thursday, November 3, 2:00-2:40

We prove that the 3-dimensional solvable group SOL is quasi-isometrically rigid, and compute is quasi-isometry group. We also prove some related results about solvable groups. This is joint work with David Fisher and Kevin Whyte.

FELSHTYN, Alexander, Szczecin University and Boise State University
Twisted Burnside theorem
Rudder 401, Friday, November 4, 4:25-4:45

This is joint work with E. Troitsky. It is proved for a wide class of groups including polycyclic and finitely generated polynomial growth groups that the Reidemeister number of a automorphism is egual to the number of finite-dimensional fixed points of induced map on the unitary dual space, if one of this number is finite.This theorem is a natural generalisation to infinite groups of the classical Burnside theorem. On the other hands it implies the congruences for Reidemeister numbers of iterations of automorphism.This congruences give necessary conditions for the realization problem of Reidemeister numbers of iterations. From another side we show that Ivanov and Osin groups give counterexamples for the twisted Burnside theorem.

FISHER, David, Indiana University
Super-rigidity and harmonic maps into infinite dimensional spaces
Rudder 301, Friday, November 4, 5:00-5:20

Let M' be an irreducible symmetric space of non-compact type that is not a real or complex hyperbolic space. Let M be a compact locally symmetric space modeled on M'. I will
discuss a complete classification of pi₁(M) equivariant harmonic maps from M' to a class of complete homogeneous Hilbert manifolds. The solution to this harmonic map problem implies a generalization of Zimmer's cocycle super-rigidity theorem and has many applications to the study of group actions on manifolds. This is joint work with Theron Hitchman.

GHYS, Étienne, École Normale Supérieure de Lyon
Geodesics on the modular surface
Rudder 301, Saturday, November 5, 3:40-4:20

The geodesic flow on the modular surface acts on SL₂(R) / SL₂(Z) which is homeomorphic to the complement ot the trefoil knot in the 3-sphere. I will try to describe the topology of this flow.

GORDON, Cameron, University of Texas
Algebraic knots with unknotting number 1
Rudder 301, Thursday, November 3, 11:50-12:30

A Conway sphere S for a knot K is a 2-sphere that meets K transversely in 4 points. S is essential if S-K is incompressible in S³ - K. Our main result is that if a knot K with an essential Conway sphere has unknotting number 1 then either the unknotting move can be isotoped to miss the canonical Bonahon-Siebenmann family of such spheres, or K belongs to the family of knots constructed by Eudave-Munoz, or K contains a tangle summand belonging to an analogous family. This gives a very strong condition on when an algebraic knot, that is not a Montesinos knot of length3, has unknotting number 1. In particular, it leads to an algorithm to determine whether or not such a knot has unknotting number 1. (Joint work with John Luecke).

GROVES, Daniel, California Institute of Technology
The Isomorphism Problem for toral relatively hyperbolic groups
Rudder 301, Thursday, November 3, 5:45-6:05

We provide a positive solution to the isomorphism problem for torsion-free groups which are hyperbolic relative to free abelian subgroups. This implies and generalises: (i) An unpublished algorithm of Gerasimov to detect if hyperbolic groups admit a free product decomposition; (ii) partially unpublished work of Sela which solves the isomorphism problem for torsion-free hyperbolic groups; (iii) recent work of Kharlampovich and Miasnikov which shows that the Grushko and JSJ decompositions of limit groups can be computed effectively; and (iv) work of Bumagin, Kharlampovich and Miasnikov which solves the isomorphism problem for limit groups. We also provide a solution to the homeomorphism problem for finite-volume
hyperbolic manifolds in dimension at least 3. (Joint work with Francois Dahmani)

GUENTNER, Erik, University of Hawaii at Manoa
Geometry of groups and approximation in group C*-algebras
Rudder 301, Saturday, November 5, 11:00-11:30

We will discuss several classes of examples in which geometric properties of discrete groups are exploited to prove that their reduced C*-algebras have approximation properties. Although a handful of properties will be mentioned, I hope to concentrate on a-T-menable groups and the completely bounded approximation property.

HERRLICH, Frank, University of Karlsruhe
A Teichmüller curve intersecting infinitely many others
Rudder 301, Friday, November 4, 4:25-4:45

This is joint work with Gabriela Schmithüsen. A torus covering that is ramified over only one point defines in a combinatorial way a special example of a complex geodesic in Teichmüller space (sometimes called ``origami''). The corresponding Veech group is always a subgroup of SL₂(Z) of finite index, and its image in the moduli space is a Teichmüller curve.

Taking a normal covering with the classical quaternion group as Galois group we obtain such an origami curve W in genus three with a lot of remarkable properties: It is one of the rare families of curves where the equation is known explicitely, and also the family of Jacobians and the (unique) cusp at the boundary of M₃. Our main result is that there are infinitely many other Teichmüller curves (which can be described combinatorially as origamis) that all intersect W. Another very interesting result, proved by M. Möller, is that W is the only Teichmüller curve that at the same time is a Shimura curve.

The origami curve W is the first and most prominent example in a whole series of origamis with Veech group SL₂(Z) that can be constructed using characteristic subgroups of the free group on two generators.

HRUSKA, Chris, University of Chicago
Cubulating relatively hyperbolic groups
Rudder 301, Friday, November 4, 5:25-5:45

We study actions of groups on CAT(0) cube complexes. This subject generalizes the classical study of group actions on trees. The existence of such an action has implications regarding codimension-1 subgroups, Property (T), a-T-menability, and biautomaticity using work of Sageev, Niblo-Roller and Niblo-Reeves.

We give criteria for determining when a relatively hyperbolic group acts on a finite dimensional cube complex and when such an action is cocompact, generalizing a theorem of Sageev from the word hyperbolic setting. More generally, we describe a ``cusped cofinite'' structure in which cube complexes for the peripheral subgroups play the role of cusps. This structure is analogous to the cusped structure of a finite volume manifold with pinched negative curvature. (Joint with Dani Wise)

ION, Patrick, Mathematical Reviews
Geometrical Fourier Transforms in the Newtonian n-Body Problem
Rudder 401, Friday, November 4, 4:00-4:20

Certain aspects of the n-body problem can be illuminated through the use of discrete Fourier transforms. The planar case serves to remind us of the usefulness of complex coordinates in elementary geometry.

KAIMANOVICH, Vadim, International University Bremen
Amenability, random walks and entropy
Rudder 301, Sunday, November 6, 2:00-2:40

The talk is devoted to a description of recent applications of random walks methods to proving amenability of certain automata groups.

KAPOVICH, Ilya, University of Illinois at Urbana-Champaign
Geodesic currents on free groups
Rudder 301, Sunday, November 6, 9:50-10:30

A geodesic current on a finitely generated free group F is a positive F-invariant measure on the set of all pairs of distinct points from the boundary of F. Geodesic currents can be thought of as measure-theoretic analogues of closed curves. We will discuss the natural "intersection form" arising in this context (and motivated by the notion of a geometric intersection number for curves on surfaces), a geometric embedding of the Culler-Vogtmann outer space in the space of projectivized currents and the new compactification of the outer space provided by this embedding.

KASSABOV, Martin, Cornell Univeristy
Small presentations of finite simple groups
Rudder 301, Saturday, November 5, 5:45-6:05

Abstract not available.

KATOK, Anatole, The Pennsylvania State University
Nets and lattices in Euclidean spaces and cocycles over the Weyl chamber flow
Rudder 301, Friday, November 4, 9:50-10:30

Abstract available here

KATOK, Svetlana, The Pennsylvania State University
Continued fractions and symbolic dynamics for the modular surface
Rudder 301, Sunday, November 6, 9:00-9:40

Geodesic flow on the modular surface can be represented as a special flow over a one-step topological Markov chain on countable alphabet using various continued fraction expansions of the end points of the geodesic at infinity and so-called "reduction theory". I will present three "standard" continued fractions methods (regular, minus, and the nearest integer continued fractions) which can be used for coding of geodesics, and outline a method of obtaining infinitely many others. I will give a dynamical interpretation of the "reduction theory" which underlines these codings via finding an attractor of a certain associated map. I will also explain how these codings are related to a more common coding of geodesics with respect to a given fundamental region.

KOBAN, Nic, Western Carolina University
The Geometric Invariant Omegaⁿ of a Product of Groups
Rudder 308, Thursday, November 3, 5:45-6:05

The invariants Omegaⁿ of a group G are analogs of the Bieri-Neumann-Strebel-Renz invariants Sigmaⁿ. For groups G and H, Omegaⁿ(GxH) is the spherical join of Omegaⁿ(G) and Omegaⁿ(H). We will discuss the proof of this statement, and we will also explore some ideas for proving the conjecture for Sigmaⁿ(GxH).

KRYLYUK, Yaroslav, Santa Fe CC, Gainesville FL
The Hessenberg tridiagonal form of the Hecke type operator of the quasi-regular representation for Grigorchuk's group
Rudder 401, Friday, November 4, 5:00-5:25

We determine the Hessenberg tridiagonal form J of the Hecke type operator of the quasi-regular representation for Grigorchuk's group G. The quasi-regular representation of G under consideration arises from the realization of this group by automorphisms on a regular binary rooted tree. The Hessenberg form J is obtained by a common "finite approximation" method through a stabilization process. We prove that J corresponds to the empiric spectral measure for G found by L. Batholdi and R.I. Grigorchuk.

KUCHMENT, Peter, Texas A&M University
Liouville theorems on abelian coverings of manifolds and graphs
Rudder 301, Sunday, November 6, 11:00-11:40

The talk will describe recent results of Y. Pinchover and the speaker concerning Liouville type theorems for equations on abelian coverings of compact Riemannian manifolds, compact complex manifolds, or finite graphs. Necessary and sufficient conditions for validity of Liouville theorems are established and exact formulas for dimensions of the spaces of polynomially growing solutions are derived. Such theorems happen to be related to the structure of the dispersion relation for the governing equation at the edges of its spectrum and require involvement of some notions from mathematical physics. At least a part of the results should hold for the case of virtually nilpotent coverings.

LIM, Seonhee, Yale University
Minimal volume entropy on graphs
Rudder 302, Friday, November 4, 5:00-5:20

For any closed Riemannian manifold of dimension at least 3, which admits a negatively curved, locally symmetric metric g, Besson, Courtois, and Gallot showed that the volume entropy is minimized (uniquely) by g among all Riemannian metrics, proving a conjecture mainly due to Gromov.

In this talk, we show the analogous problem for graphs: Among the normalized metrics on a finite graph, we show the existence and the uniqueness of an entropy-minimizing metric, and give explicit formulas for the minimal volume entropy and the metric realizing it.

MATUCCI, Francesco, Cornell University
The Simultaneous Conjugacy Problem for Thompson's Group F is solvable
Rudder 501, Thursday, November 3, 5:20-5:40

I will discuss the an alternative approach to the Conjugacy Problem Thompson's group F = PL₂(I) which, unlike the usual approach using diagram groups, is also applicable to the group of piecewise linear orientation-preserving homeomorphisms of the unit interval with a finite number of breakpoints (often denoted PL₀(I)). This can be used to solve the simultaneous conjugacy problem.

MORRIS, Dave Witte, University of Lethbridge
Some arithmetic groups that cannot act on the line
Rudder 301, Thursday, November 3, 5:20-5:40

It is known that finite-index subgroups of the arithmetic group SL₃(Z) have no nontrivial actions on the real line. This naturally led to the conjecture that most other arithmetic groups (of higher real rank) also cannot act on the line. This problem remains open, but joint work with Lucy Lifschitz verifies the conjecture for many examples. This includes all finite-index subgroups of SL₂(Z[a]), where a is the square root of any square-free integer greater than 1. The proofs are based on the fact, proved by D.Carter, G.Keller, and E.Paige, that every element of these groups is a product of a bounded number of elementary matrices.

MOZES, Shahar, Hebrew University of Jerusalem
Automata and square complexes
Rudder 301, Friday, November 4, 2:00-2:40

Abstract not available

NEKRASHEVYCH, Volodymyr, Texas A&M University
Thurston equivalence of topological polynomials
Rudder 301, Saturday, November 5, 1:40-2:20

We will describe how to decide when two topological polynomials are conjugate up to homotopies. This will answer some open questions in holomorphic dynamics (in particular the "twisted rabbit" question by J. Hubbard). This is a joint work with L.Bartholdi.

NIBLO, Graham, University of Southampton
Minimal cubings and intersection numbers
Rudder 301, Friday, November 4, 4:00-4:20

Sageev associated a CAT(0) cubing to any subgroup H of a group G such that the number of ends of the pair (G,H) is at least 2. The cubing is not canonical. We discuss a variation on Sageev's construction which allows us to produce a new cubing more naturally associated to the pair. The construction is different from but related to Guidardel's convex core for an action on a product of trees and can yield information about algebraic intersection numbers. This is joint work with Sageev, Scott and Swarup.

NIKOLOVA-POPOVA, Daniela, Bulgarian Academy of Sciences
On the solubility of finite groups satisfying certain two-variable commutator identities
Rudder 401, Friday, November 4, 5:25-5:45

It is a well known fact that a final group is nilpotent iff it satisfies an Engel law. Our aim is to find a similar characterization for finite soluble groups. For this purpose, the computer algebra system GAP is used for the calculation of some matrix eqautions in finite simple groups. The final goal is to obtain a characterization of the radical of solubility.

OL'SHANSKII, Alexander, Vanderbilt University
Dehn functions of groups: between quadratic and nonrecursive
Rudder 310, Thursday, November 3, 9:50-10:30

I will mainly speak of joint results with Mark Sapir on the behavior of group Dehn ( = isoperimetric) functions. In particular, we have constructed a finitely presented group (1) with undecidable word problem, and so its Dehn function f(n) is not bounded from above by any recursive function, but (2) f(n) is bounded by a quadratic function on arbitrary long intervals.

OSIN, Denis, City University of New York
Peripheral fillings of relatively hyperbolic groups
Rudder 301, Thursday, November 3, 4:00-4:30

A group theoretic version of Dehn surgery is studied. Starting with an arbitrary relatively hyperbolic group G we define a peripheral filling procedure, which produces quotients of G by imitating the effect of the Dehn filling of a complete finite volume hyperbolic 3-manifold on its fundamental group. The main result is an algebraic analogue of Thurston's hyperbolic Dehn surgery theorem. We also show that peripheral subgroups of G "almost" have the Congruence Extension Property. Various geometric and algebraic applications of these results will be discussed.

PIGGOTT, Adam, Tufts University
Groups Ends and the Todd--Coxeter Procedure
Rudder 501, Saturday, November 5, 5:20-5:40

We discuss the manifestation of group ends in the Todd--Coxeter Procedure.

PILGRIM, Kevin, Indiana University
Uniformly quasiconformal dynamical systems are ubiquitous
Rudder 301, Saturday, November 5, 11:40-12:10

We give general, robust, purely topological conditions on a topological dynamical system f: X -> X to admit a metric for which the iterates of f are uniformly quasiconformal. Moreover, this metric is unique up to quasisymmetry. The construction uses techniques from the theory of Gromov hyperbolic spaces. As an application, we obtain new numerical topological invariants of certain dynamical systems.

PITTET, Christophe, Université de Provence Aix-Marseille 1
Random walks on locally compact groups
Rudder 301, Sunday, November 6, 2:50-3:30

If G is a compactly generated group and H is a cocompact lattice in G then the return probabilities of random walks on G and on H are comparable.

POPA, Sorin, University of California at Los Angeles
Superrigidity of Bernoulli actions
Rudder 301, Saturday, November 5, 9:00-9:40

During the early 80's Zimmer proved that if G is a higher rank lattice and H is a simple linear algebraic group then any H-valued cocycle of a free ergodic measure preserving G-action satisfying certain ``non-degeneracy'' conditions is cohomologous to a group morphism of G into H. Such "cocycle superrigidity" can be used to prove orbit equivalence (OE) rigidity results for actions, through an approach initiated by Zimmer and a series of techniques developed by Furman.

We will show that cocycle superrigidity holds whenever G has an infinite normal subgroup with the relative property (T) of Kazhdan-Margulis (e.g. G an infinite Kazhdan group), H is a closed subgroup of the unitary group of a finite von Neumann algebra (e.g. H discrete), and the G-action is Bernoulli. The proof uses "deformation/rigidity" techniques in von Neumann algebra framework. We use this same framework to deduce ``OE superrigidity'' from ``cocycle superrigidity'' in a new manner, giving alternative proofs to our OE results in math.OA/0407103, with sharp generalizations.

RADIN, Charles, University of Texas
The geometric symmetry of dense packings
Rudder 301, Saturday, November 5, 5:20-5:40

It is a classic open problem to understand why densest packings tend to have crystallographic symmetry. We discuss the extension of this phenomenon to packings which are merely very dense, using both simulations and proofs.

REID, Alan, University of Texas
Large covers of arithmetic 3-manifolds
Rudder 301, Saturday, November 5, 2:30-3:10

We will discuss recent progress on the problem of showing that every arithmetic 3-manifold has a finite cover whose fundamental group surjects a free non-abelian group.

SABALKA, Lucas, University of Illinois at Urbana-Champaign
A rigidity result for tree braid groups
Rudder 308, Thursday, November 3, 5:20-5:40

Given a tree braid group B_nT on n = 4 strands, we are able to reconstruct the tree T. Thus tree braid groups B₄T and trees T (up to homeomorphism) are in bijective correspondence. There is much hope that the result holds for any n >3. That such a bijection exists is not true for braid groups on any spaces of dimension more than 1. We believe the bijection may be applied to solve the isomorphism problem for tree braid groups.

SAEED, Muhammad Sarwar, Heriot-Watt University
Magnus Subgroups in One-Relator Surface Groups
Rudder 501, Saturday, November 5, 5:45-6:05

Let S be a closed, connected and orientable surface. A one-relator surface group is the quotient of the fundamental group of S by the normal closure of a single element. Let G=(X,R) be a one-relator surface group. A magnus subgroup of G is a subgroup M generated by a proper subset L of X where L omits at least one generator occurring in R. We shall prove that Magnus subgroups of one-relator surface groups are cyclonormal.

SALOFF-COSTE, Laurent, Cornell University
Lie groups and Property RD
Rudder 301, Sunday, November 6, 11:50-12:30

A locally compact group equipped with a length function has property RD if there is a polynomial P(R) such that the norm of the convolution operator associated with a continuous function f supported in a ball of radius R is bounded by P(R) times the L² norm of f. About 30 years ago, Haagerup discovered that the free group on k generators has property RD. Hyperbolic groups (in the sense of Gromov) have property RD (P. de la Harpe). A. Valette conjectures that all cocompact lattices in semisimple Lie groups have property RD. This talk will describe the main result of a joint work with Indira Chatterji and Christophe Pittet which characterizes the connected Lie groups that have property RD.

SAPIR, Mark, Vanderbilt University
Groups acting on tree-graded spaces
Rudder 301, Thursday, November 3, 11:00-11:40

I am going to talk about our joint work with Cornelia Drutu on groups acting on tree-graded spaces, and applications to relatively hyperbolic groups.

SCHMITHÜSEN, Gabriela, University of Karlsruhe
Congruence and non congruence groups that are Veech groups of special Teichmüller curves
Rudder 308, Friday, November 4, 5:00-5:20

In this talk I would like to show that most congruence subgroups of SL₂(Z) occur as Veech groups for a Teichmüller curve. I shall also present certain examples for Veech groups that are non congruence groups.

A Teichmüller curve is in general defined by a Riemann surface X together with a quadratic differential. The latter fixes a flat structure on X. This additional structure assigns a certain discrete subgroup of SL₂(R) to the surface, called the Veech group. The Veech group is known to (almost) determine the Teichmüller curve. I study particular Teichmüller curves, sometimes called origami curves: They are defined by a differential, which is the pull back of a covering from X to an elliptic curve, ramified only over one point. In this case the Veech group is a subgroup of SL₂(Z). Using the natural projection from the automorphism group of the free group F₂ on two generators to SL₂(Z), one has the following characterisation for these Veech groups: They are precisely the images of subgroups G(U) of Aut(F₂), which stabilize a finite index subgroup U of F₂. This leads naturally to the question: Which subgroups of SL₂(Z) are images of such stabilizing groups?

My main result is that each congruence group of arbitrary level n that is the stabilizing group of its own orbit space on (Z/nZ)² has this property. In fact most congruence groups occur this way. For example all congruence groups of prime level are of this type with five exceptions for small primes. Furthermore, I will give examples for Veech groups that are non congruence groups and present a method to construct an infinite sequence of such Veech groups. In particular, I will use this to find such an example in each moduli space M_g (g > 1) .

SHLYAKHTENKO, Dimitri, University of California at Los Angeles
Free entropy dimension for groups
Rudder 301, Saturday, November 5, 9:50-10:30

We discuss joint work with A. Connes and joint work with I. Mineyev which leads to the computation of Voiculescu's non-microstates free entropy dimension for generators of group algebras in terms of the L² cohomology of the group.

SMIRNOVA-NAGNIBEDA, Tatiana, Université de Genève
Boundary measures on metric trees and study of the Outer space
Rudder 301, Friday, November 4, 11:00-11:40

We discuss Hausdorff measures on boundaries of uniform metric trees and geodesic currents on free groups associated with them. As a result we construct a compactification of the Outer Space and solve the minimal volume entropy problem in the Outer Space setting.

SONKIN, Dmitriy, University of Virginia
Compression of uniform embeddings into Hilbert space
Rudder 308, Friday, November 4, 4:25-4:45

If one tries to embed a metric space uniformly into Hilbert space, how close to quasi-isometric could the embedding be? We discuss this question for hyperbolic groups and for finite dimensional CAT(0) cube complexes. In particular, we show that the Hilbert space compression of any hyperbolic group is 1.

STALDER, Yves, Université de Neuchâtel
Limits of Baumslag-Solitar groups
Rudder 302, Friday, November 4, 4:25-4:45

We are interested in the limits of the (marked) Baumslag-Soliatr groups BS(m,n) = < a,b | ab^ma^-1 = bⁿ > in the space of marked groups endowed with the Cayley-Grigorchuk topology. I proved that BS(1,n) --> Z wr Z and that the sequence (BS(m,n))_n does not converge if |m|>1. But it is possible to associate a limit (of some subsequence) to any m-adic integer. In the talk, I will define the limits, discuss some of their properties, in particular a-T-menability, and give presentations of them. This is joint work with Luc Guyot.

THOMAS, Anne, University of Chicago
Lattices in automorphism groups of hyperbolic buildings
Rudder 302, Saturday, November 5, 5:20-5:40

Let G be a locally compact group and let Gamma be a lattice in G, that is, a discrete subgroup of cofinite volume. For example, the classical setting for questions about lattices is where G is an algebraic group. More recently, the lattices in G the automorphism group of a tree have been studied. We consider the lattices in G the automorphism group of a higher-dimensional polyhedral complex, such as a hyperbolic building. In particular, when G is the automorphism group of Bourdon's building I_pq, we find the exact set of covolumes of cocompact lattices, and show that G admits an infinite ascending tower of cocompact lattices.

VOGTMANN, Karen, Cornell University
Structure at infinity of Outer space
Rudder 301, Friday, November 4, 11:50-12:30

The group Out(F_n) of outer automorphisms of a free group acts with finite stabilizers on a geometric object known as Outer space. This action is not cocompact, but Bestvina and Feighn showed how to bordify Outer space by adding structure at infinity, so that the action extends and the quotient is compact. I will describe this and show how all of the known rational cohomology of Out(F_n) is supported "at infinity" in the quotient of this bordification.

VOROBETS, Yaroslav, Clay Mathematics Institute
Free groups defined by finite automata
Rudder 301, Friday, November 4, 2:50-3:30

An invertible initial automaton canonically defines an automorphism of a regular rooted tree. An invertible finite noninitial automaton defines a finitely generated group of such automorphisms.
We present a series of finite automata, each defining a free group of rooted tree automorphisms. These are the first examples of the kind.

WALSH, Genevieve, University of Texas
Surfaces in finite covers and the group determinant
Rudder 501, Saturday, November 5, 5:45-6:05

We show that infinitely many fillings of a 1-cusped hypebolic 3-manifold are virtually Haken. The main tool is the group determinant of the group of covering transformations of a finite regular cover. Joint with Daryl Cooper.

WIENHARD, Anna, Institute for Advanced Study, Princeton
Symplectic Anosov structures on Riemann surfaces
Rudder 308, Friday, November 4, 4:00-4:20

The Toledo invariant is associated to a representation of the fundamental group of a closed surface of higher genus into the symplectic group. We study representations with maximal Toledo invariant and associate to every such representation an Anosov structure on the unit tangen bundle. As a consequence we prove that any such representation is fairhful with discrete image. The limit set of the image is a rectifiable circle and the representation is a quasi-isometric embedding. This is joint work with Marc Burger, Alessandra Iozzi and Francois Labourie.

XIE, Xiangdong, University of Cincinnati
Uniform exponential growth of relatively hyperbolic groups
Rudder 302, Friday, November 4, 5:25-5:45

A relatively hyperbolic group has uniform exponential growth unless it is virtually infinite cyclic.

YOUNG, Robert, University of Chicago
Averaged Dehn Functions for Nilpotent Groups
Rudder 302, Thursday, November 3, 5:45-6:05

Gromov proposed an averaged version of the Dehn function and claimed that in many cases it should be subasymptotic to the Dehn function. Using results on random walks in nilpotent groups, we give upper and lower bounds for a version of this averaged Dehn function which confirm this claim for finitely-generated torsion-free nilpotent groups. In particular, if such a group satisfies the isoperimetric inequality FA(n) < Cn^a for a>2 then it satisfies the averaged isoperimetric inequality FA_avg(n) < C'n^a/2. In the case of non-abelian free nilpotent groups, the bounds we give are asymptotically sharp.

ZELMANOV, Efim, University of California at San Diego
Asymptotic properties of infinite algebras
Rudder 301, Friday, November 4, 9:00-9:40

We will discuss (i) some very old problems concerning nil algebras of slow (polynomial) growth and their relations to branch groups and algebras, and (ii) very fast growing algebras and their relations to groups and algebras with property tau.