Organizers: Bruce Hughes and Mark Sapir

Wednesdays, 4:10pm in SC 1424 (unless otherwise noted)

• Date: 21 Jan 2004
  • Speaker: Victor Guba, Vologda State Pedagogical Institute
  • Title:  On the properties of the Cayley graph of R.Thompson's group F
  • Abstract:
    In this talk, we review recent results that concern the amenability
     problem for R.Thompson's group F and the problem about the growth
     function of this group. We show how to construct finite subgraphs
     in the Cayley graph with high "density," present some lower bounds
     for the growth function and give an easy algorithm to find the length
     of an element of F in its natural generators.

• Date: 28 Jan 2004
  • Speaker: Mark Sapir, Vanderbilt University
  • Title: Polynomial maps over finite fields and residual finiteness of mapping tori
  • Abstract:
    (with Alexander Borisov)
    We prove that any ascending HNN extension of a linear group is residually
    finite (for non-linear groups there are counterexamples). This result and
    some facts about random walks on the plane imply that most 1-related groups
    are residually finite. The key point in the proof consists of counting the
    number of quasi-fixed points of a polynomial map over a field of prime
    characteristic. In particular, we prove that the set of quasi-fixed points
    of any dominant polynomial map of an affine scheme over a finite field is
    Zariski dense.
    
     
• Date: 4 Feb 2004
  • Speaker: Mark Sapir, Vanderbilt University
  • Title: Polynomial maps over finite fields and residual finiteness of mapping tori, continued
  
   

• Date:  11 Feb 2004
  • Speaker:  John Ratcliffe, Vanderbilt University
  • Title: Some examples of aspherical 4-manifolds that are homology 4-spheres
  • Abstract: (with Steve Tschantz) Some examples of aspherical 4-manifolds that are homology 4-spheres will be described.  These are the first examples of such manifolds and their existence answers a 1977 question of William Thurston.
     

   

• Date: 19 Feb 2004 (Thursday) Mathematics Colloquium at 4:10pm in SC 1206
  • Speaker: Frank Quinn, VPI & SU
  • Title: What is a manifold?
  • Abstract: Short names such as "manifold" or "group" tend to be attached to central objects in an area, as identified by tradeoffs between generality and properties. More general objects with significantly fewer properties, such as "homology manifolds" or "semigroup", and more special objects with slightly more properties, such as "real analytic manifold" or "algebraic group" are thought of as perturbations on the central object. However what we see as the "central object" often changes as we learn more. In this lecture I'll trace changes in the meaning of "manifold" as it tracked development of the subject over the last century. We may be on the verge of another major change of viewpoint, though whether the word "manifold" will follow it this time remains to be seen.

• Date: 20 Feb 2004 (Friday) at 4:10 in SC 1424
  • Speaker: Frank Quinn, VPI & SU
  • Title: Controlled algebraic K-theory
  • Abstract: Controlled algebra was developed to formulate obstructions to topological problems. Some of the obstructions could be expressed in terms of K-theory, and this was first regarded as a "computation" of obstructions. Farrell, Hsiang, Jones and others then showed how to  use the process backwards: use special properties of the topological problems to get new information about K-theory. Controlled K-theory streamlines this by replacing "topological problems with certain obstructions" as the controlled side of the argument. This is used to prove cases of an "infinite hyperelementary induction" conjecture that generalizes the Farrell-Jones "fibered isomorphism conjecture".
     

• Date: 25 Feb 2004
  • Speaker: O.V.Kulikova, Moscow State University
  • Title: On intersections of normal subgroups in free groups
  • Abstract:   It is well-known that any group can be
    presented as a factor-group of a free group $F$ by a normal
    subgroup. Let $G_1$ (respectively $G_2$) be a factor-group of $F$
    by the normal closure $N_1$ (respectively $N_2$) of a set $R_1$
    (respectively $R_2$) of cyclically reduced words of $F$. Then
    there is a natural homomorphism from $F$ to the direct product
    $G_1 \times G_2$ with the kernel equal to the intersection $N_1
    \cap N_2$. There is a natural question to describe this kernel. In
    the talk we will give necessary and sufficient conditions on $R_1$
    and $R_2$ when the kernel is equal to the mutual commutant $[ N_1,
    N_2 ]$. These conditions will be expressed in terms of certain
    geometric objects called pictures.

   

• Date: 19 March 2004 (Friday) at 4:10 in SC 1424
  • Speaker: Goulnara Arjantseva, University of Geneva
  • Title: Spectral characterization of property T for finitely generated groups after Gromov
  • Abstract: A sufficient condition is given which guarantees that a discrete group has Kazhdan's property T.
    The presentation is based on works of Ghys, Gromov, Valette, Zuk.
     

• Date: 24 March 2004
  • Speaker:  Cornelia Drutu, University of Lille-1
  • Title: Tree-graded spaces and asymptotic cones of groups
  • Abstract:
    The talk is on a joint work with M. Sapir. We introduce a concept of
    tree-graded metric spaces and use it to show quasi-isometry invariance of
    certain classes of relatively hyperbolic groups, to obtain a
    characterization of relatively hyperbolic groups in terms of their
    asymptotic cones, to find geometric properties of Cayley graphs of
    relatively hyperbolic groups, and to construct (using also ideas of
    Olshanskii, Erschler and Osin) the first example of a finitely generated
    group with continuum non-quasi-isoimetric asymptotic cones. Note that by a
    result of Kramer, Shelah, Tent, and Thomas, continuum is the maximal
    possible number of different asymptotic cones provided the Continuum Hypothesis is true.

   

• Date: 31 March 2004
  • Speaker: Cornelia Drutu, University of Lille-1
  • Title:  Tree-graded spaces and asymptotic cones of groups (continued)
  • Abstract:
    The talk is on a joint work with M. Sapir. We introduce a concept of
    tree-graded metric spaces and use it to show quasi-isometry invariance of
    certain classes of relatively hyperbolic groups, to obtain a
    characterization of relatively hyperbolic groups in terms of their
    asymptotic cones, to find geometric properties of Cayley graphs of
    relatively hyperbolic groups, and to construct (using also ideas of
    Olshanskii, Erschler and Osin) the first example of a finitely generated
    group with continuum non-quasi-isoimetric asymptotic cones. Note that by a
    result of Kramer, Shelah, Tent, and Thomas, continuum is the maximal
    possible number of different asymptotic cones provided the Continuum Hypothesis is true.

• Date: 7 April 2004
  • Speaker: Maria Milentyeva, Moscow State University
  • Title: On The Torsion-free ranks of finitely generated nilpotent groups and of their abelian subgroups
  • Abstract: In 1982 John Wilson raised question about the relationship between the torsion-free rank of a finitely generated nilpotent group and the ranks of its abelian subgroups. This question was put in the following more precise form in the 1992 edition of the Kourovka Notebook: "Give a realistic upper bound for the torsion  free rank of a finitely generated nilpotent group in terms of the ranks of its abelian subgroups. More precisely, for any integer n we denote by f(n) the greatest integer h  such that there exists a  finitely generated nilpotent group of   torsion free rank  h such that ranks without torsion of all abelian subgroups of this group are not greater than n. It is easy to see that f(n) <=n(n+1)/2.   Describe the asymptotic behavior of f(n) as n tends to infinity. Is f(n)  bounded  below by a  non-linear quadratic function of n?" We will show that this final question has an affirmative answer and the function f(n) satisfies the following inequality:   f(n) >=(n*n-4)/8+n.

   

• Date: 14 April 2004
  • Speaker: Denis Osin, Vanderbilt University
  • Title: Relatively hyperbolic groups and embedding theorems
  • Abstract: In this talk we will discuss a method of proving embedding theorems for countable groups based on the notion of relative hyperbolicity. For example, we show that every torsion--free countable group can be embedded into a 2--generated torsion--free group with exactly 2 counjugacy classes. In particular this gives the affirmative answer to the well known question of the existence of a finitely generated torsion--free group G other than Z/2Z such that all non--trivial elements are conjugate in G.
     

   

• Date: 21 April 2004
  • Speaker: Pedro Ontaneda, Binghamton University and UFPE, Recife, Brazil
  • Title: Ricci flow and pinched negative curvature
  • Abstract: I will discuss the smooth convergence of the Ricci flow starting with
    a pinched negatively curved Riemannian metric.