Organizers: Bruce Hughes and Mark Sapir

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

• Date: 1 September 2004
  • Speaker: Mark Sapir, Vanderbilt University
  • Title:  Rapid decay and relatively hyperbolic groups
  • Abstract:
    This is a joint work with Cornelia Drutu. We prove that a relatively
    hyperbolic group satisfies RD if and only if its parabolic subgroups satisfy
    RD. In fact relative hyperbolicity in this statement can be replaced by a
    natural weak geometric condition.

• Date: 8 September 2004
  • Speaker: Alexander Ol'shanskii, Vanderbilt University
  • Title: Groups with almost quadratic Dehn function
  • Abstract:
    This is a joint work with Mark Sapir.  We construct groups whose Dehn

    functions are equivalent to  $n^2\log n$. There are groups with undecidable

    conjugacy problem among them. (The conjugacy problem is decidable if

    the Dehn function is subquadratic.) There are other examples. In particular,

    there exists a group having Dehn function $f$ with $f(n_i) \le  cn_i ^2 $ for

    a constant $c$ and a sequence $n_i \too\infty$,  but $f(m_i) / m_i ^2 \too \infty$

    for another sequence $m_i \too \infty$.

• Date: 13 October 2004
  • Speaker: Denis Osin, Vanderbilt University
  • Title:  Congruence Extension Property and relative hyperbolicity
  • Abstract: A subgroup H of a group G is said to have the Congruence Extension Property (CEP) if any normal subgroup of H is an intersection of H and a normal subgroup of G. Clearly H has CEP whenever H is a free factor of G. We generalize this result to the situation when G is hyperbolic relative to H. This allows to obtain several new embedding--type theorems for countable groups. For example, we show that any non--elementary group hyperbolic with respect to a collection of proper subgroups is SQ--universal. Among other results, we give answers to questions 7.54 and 7.56 from Kourovka notebook.

   

• Date: 20 October 2004
  • Speaker: Sorin Popa, University of California at Los Angeles
  • Title: Some Calculations of 1-Cohomology for Actions of Groups
  • Abstract: For each group G having an infinite normal subgroup with relative property (T) and each countable abelian group Г we construct measure-preserving actions σ_Г of G on the probability space such that the 1'st cohomology group of σ_Г, H¹(σ_Г,G), is equal to Char(G)×Г. We deduce that G has uncountably many non stably orbit equivalent actions. We also calculate 1-cohomology groups and show existence of ``many'' non stably orbit equivalent actions for free products of groups as above.
     

   

• Date: 27 October 2004
  • Speaker: Denis Osin, Vanderbilt University
  • Title: Congruence Extension Property and relative hyperbolicity, continued
  • Abstract: A subgroup H of a group G is said to have the Congruence Extension Property (CEP) if any normal subgroup of H is an intersection of H and a normal subgroup of G. Clearly H has CEP whenever H is a free factor of G. We generalize this result to the situation when G is hyperbolic relative to H. This allows to obtain several new embedding--type theorems for countable groups. For example, we show that any non--elementary group hyperbolic with respect to a collection of proper subgroups is SQ--universal. Among other results, we give answers to questions 7.54 and 7.56 from Kourovka notebook.

   

• Date: 3 November 2004
  • Speaker: Greg Friedman, Yale University
  • Title: Singular chains in intersection homology
  • Abstract:
    Intersection homology is a generalization of standard homology particularly well-suited to the study of singular spaces, spaces that are not quite manifolds. For example intersection homology allows one to extend Poincare duality to complex algebraic varieties. I will provide an overview of intersection homology, focusing on the two very different methods of defining it - geometric chains versus sheaf theory - and my own recent work to reconnect the two approaches.