Organizers: Bruce Hughes and Mark Sapir

Wednesdays, 4:10pm in SC 1307 (unless otherwise noted)

• Date: 14 September 2005
  • Speaker:  Greg Friedman, Vanderbilt University
  • Title: Intersection homology of stratified fibrations and neighborhoods
  • Abstract: I will discuss my recent work on computing the intersection homology
    of approximate tubular neighborhoods of pure subsets of manifold stratified spaces.
    These neighborhoods generalize the regular and tubular neighborhoods of simplicial
    or smooth topology to the category of manifold stratified spaces.

• Date: 21 September 2005
  • Speaker: Dan Guralink, Vanderbilt University
  • Title: Coarse decompositions of boundaries for CAT(0)-groups
  • Abstract:
    In this talk we shall introduce a combinatorial notion of boundary $\Re H$
    for a cubing arising as the dual of a discrete $\omega$-dimensional
    poc-set $H$. $\Re H$ has a natural median algebra structure, as well as a 
    natural ordering whose intervals coincide with its intervals as a median
    algebra.
    When $H$ happens to be a (discrete) $G$-invariant system of halfspaces in
    a CAT(0) space $X$ endowed with a geometric action by a group $G$, we
    show how one can use $\Re H$ for producing a topologically meaningful 
    decomposition/stratification of the CAT(0) boundary $\partial X$ of $X$,
    having much to do with both the cone and Tits topologies on $\bd X$.
    The sets into which $\bd X$ is decomposed are defined as the fibers of a
    (discontinuous) map $\rho$ of $\bd X$ into $\Re H$; this map is
    actually well-defined for \emph{any} reasonable halfspace system in $X$,
    but has more interesting properties in the presence of a $G$-action.
    Finally, we provide a criterion (in terms of the image of $\rho$) for $G$
    to act co-compactly on the cubing dual to $H$.
    Our constructions and results provide a setting in which several issues of
    interest to geometric group theory are tied together: the end structure of
    semi-splittable groups, CAT(0) boundary topology, co-compact cubulations.
    In view of the results by Niblo-Reeves, Williams and Caprace, Coxeter
    groups supply a particularly good example of a setting to which this machinery
    can be applied in hopes of understanding their CAT(0) boundaries and the
    connections between boundary topology and, say, visual splittings of the
    corresponding group.

• Date: 28 September 2005
  • Speaker: Stefan Forcey, Tennessee State University
  • Title:  Higher categories and geometric combinatorics of free groups
  • Abstract:
    This talk will discuss a pair of projects in the area of  connecting homotopy
    coherence with categorical coherence. First is the problem of finding a complete
    combinatorial description of two new sequences of convex polytopes which have
    important universal properties. They are among the descendents of Stasheff's
    associahedra, the parameterizing operad for homotopy associative products
    in A_n spaces. In a comparison of vertex  data in the underlying graphs of the
    associahedra, composihedra and naturahedra the language of a free monoid
    on a collection of letters is convenient. The free group case leads to interesting
    generalizations. The application of these polytopes is to the definition of categories
    weakly enriched over a monoidal n-category.

   

• Date: 5 October 2005
  • Speaker: Daniela Nikolova, FAU and Bulgarian Academy of Sciences
  • Title: Characterisation of Finite Soluble Groups by Two-Variable
    Commutator Identities
  • Abstract: There are different approaches concerning the characterization of
    soluble groups. Some of them are generic. We use here another approach
    related to the study of the identities in finite soluble groups.
    The solubility is checked by examining a counter-example of least
    possible order, i.e. we attempt to show that in minimal simple groups there
    are no such identities. Thus, we solve equations in matrix groups,
    especially for the series of groups G = PSL(2, p), involving computer
    calculations.
    The solution of such equations is connected to some well-known problems
    in finite matrix groups, as for example the Ore conjecture.

   

• Date: 12 October 2005
  • Speaker: Dan Guralink, Vanderbilt University
  • Title: Coarse decompositions of boundaries for CAT(0)-groups (continued)
  • Abstract:
    In this talk we shall introduce a combinatorial notion of boundary $\Re H$
    for a cubing arising as the dual of a discrete $\omega$-dimensional
    poc-set $H$. $\Re H$ has a natural median algebra structure, as well as a 
    natural ordering whose intervals coincide with its intervals as a median
    algebra.
    When $H$ happens to be a (discrete) $G$-invariant system of halfspaces in
    a CAT(0) space $X$ endowed with a geometric action by a group $G$, we
    show how one can use $\Re H$ for producing a topologically meaningful 
    decomposition/stratification of the CAT(0) boundary $\partial X$ of $X$,
    having much to do with both the cone and Tits topologies on $\bd X$.
    The sets into which $\bd X$ is decomposed are defined as the fibers of a
    (discontinuous) map $\rho$ of $\bd X$ into $\Re H$; this map is
    actually well-defined for \emph{any} reasonable halfspace system in $X$,
    but has more interesting properties in the presence of a $G$-action.
    Finally, we provide a criterion (in terms of the image of $\rho$) for $G$
    to act co-compactly on the cubing dual to $H$.
    Our constructions and results provide a setting in which several issues of
    interest to geometric group theory are tied together: the end structure of
    semi-splittable groups, CAT(0) boundary topology, co-compact cubulations.
    In view of the results by Niblo-Reeves, Williams and Caprace, Coxeter
    groups supply a particularly good example of a setting to which this machinery
    can be applied in hopes of understanding their CAT(0) boundaries and the
    connections between boundary topology and, say, visual splittings of the
    corresponding group.

• Date: 19 October 2005
  • Speaker: Mark Sapir, Vanderbilt University
  • Title:  Groups acting on tree-graded spaces and automorphisms of relatively hyperbolic groups
  • Abstract: This is a joint work with Cornelia Drutu. We develop a theory of groups acting on tree-graded
    spaces generalizing the Rips-Sela-Bestvina-Feighn theory of groups acting on R-trees. As a result, we
    describe relatively hyperbolic groups with infinite automorphism groups, co-Hopfian relatively hyperbolic
    groups, etc.

   

• Date: 26 October 2005
  • Speaker: Mark Sapir, Vanderbilt University
  • Title:  Groups acting on tree-graded spaces and automorphisms of relatively hyperbolic groups
    (continued)
  • Abstract: This is a joint work with Cornelia Drutu. We develop a theory of groups acting on tree-graded
    spaces generalizing the Rips-Sela-Bestvina-Feighn theory of groups acting on R-trees. As a result, we
    describe relatively hyperbolic groups with infinite automorphism groups, co-Hopfian relatively hyperbolic
    groups, etc.

   

• Date: 2 November 2005
  • Speaker: Dan Guralink, Vanderbilt Univeristy
  • Title: Ends of relatively-hyperbolic groups and circle packings
  • Abstract:
    In 1996, Bowditch produced the JSJ splitting of a one-ended
    word-hyperbolic group G using only topological and dynamical information
    about the Gromov boundary X= ∂G. Using the fact that G is a
    uniform discrete convergence group of X, Bowditch constructed a tree T
    and an action of G on T that is non-trivial whenver T is non-empty.
    In this talk we discuss a generalization of his techniques to the wider
    setting of minimal cusp-uniform (AKA geometrically finite) discrete
    convergence group actions. We will show on the example of the Apollonian
    packing how to construct a tree that proves that any minimal cusp-uniform
    group of homeomorphisms of X is virtually free.
    More generally, for simplicity we will consider only spaces X having
    some common characteristics with circle-packings on the sphere: X is a
    locally-connected continuum having no global cut point, no point of
    infinite valence, and no cut-pairs.
    Time permitting, we will discuss how to show that if such a space X
    admits a minimal cusp-uniform action by a group G and obeys a certain
    local planarity condition, then X is homeomorphic to a circle packing
    on the sphere.

     

• Date: 9 November 2005
  • Speaker: Dušan Repovš, University of Ljubljana
  • Title: New results on wild, homogeneous and rigid Cantor sets in Euclidean 3-space
  • Abstract: The first part of the talk will be a historical survey on wild Cantor sets in R³, the first such
    set being constructed by Louis Antoine already in the 1920's in his Dissertation, after he was blinded while
    serving in the French army during WWI. In the main part of the talk we shall
    present a new general technique for constructing wild Cantor sets in R³ which are nevertheless
    Lipschitz homogeneously embedded into R³. Applying the well-known Kauffman version of the Jones polynomial
    we shall show that our construction produces even uncountably many topologically inequivalent wild
    Cantor sets in R³. These Cantor sets have the same number of components in the interior of each
    stage of the defining sequence and are Lipschitz homogenous. We shall also present construction of
    rigid wild Cantor sets in R³ with simply connected complement. In conclusion, we plan to state
    some open problems and conjectures.

• Date: 16 November 2005
  • Speaker: Greg Friedman, Vanderbilt University
  • Title: A gentle introduction to intersection homology

• Date: 2 December 2005 (Friday), Room SC 1307, 4:10 - 5:00
  • Speaker: Volodymyr Nekrashevych, Texas A&M University
  • Title: An uncountable family of 3-generated groups
  • Abstract:
    We will present a family of 3-generated groups G_w acting on the binary
    rooted tree, parametrized by infinite binary sequences w. We will show
    that the isomorphism classes of the family are countable, thus we get
    uncountably many different groups. On the other hand, we will show that
    for any finite number of relations between the generators of any group of
    the family we can find in any other group G_w a generating set for which
    the same relations hold. We will also describe (infinite) presentations of
    each of the groups G_w and show a relation of the family with iterations
    of a two-dimensional rational map. Several open problems will be
    discussed.

• Date: 7 December 2005
  • Speaker: Greg Friedman, Vanderbilt University
  • Title: A gentle introduction to intersection homology (continued)