Geometry/Topology Seminar

Winter 2008

Thursdays (and sometimes Tuesdays) 2-3pm, in Ryerson 358

- Thursday January 10 at 2pm in Ry358
- Nathan Broaddus, University of Chicago
- Homology of the curve complex and the Steinberg module of the mapping class group
- Abstract: The homology of the curve complex is of fundamental importance for the homology of the mapping class group. It was previously known to be an infinitely generated free abelian group, but to date, its structure as a mapping class group module has gone unexplored. I will give a resolution for the homology of the curve complex as a mapping class group module. From the presentation coming from the last two terms of this resolution I will show that this module is cyclic and give an explicit single generator. As a corollary, this generator is a homologically nontrivial sphere in the curve complex.
- Thursday January 17 at 2pm in Ry358
- Mustafa Korkmaz, Middle East Technical University
- The curve complex on a nonorientable surface
- Abstract: The study of the mapping class group of an orientable surface via its action on the curve complex was initiated by Nikolai Ivanov, who proved that for most orientable surfaces, the natural map from the mapping class group to the automorphism group of the curve complex is an isomorphism. This remarkable result has been extended in various directions since then. In this talk, I will give a proof of the nonorientable version of this result, which is a joint work with Ferihe Atalan
- Thursday January 24 at 2pm in Ry358
- Benson Farb, University of Chicago
- Analogies and contrasts between Riemann's moduli space and locally symmetric spaces
- Abstract: It is a classical theme to compare and contrast the moduli space of genus g Riemann surfaces with locally symmetric orbifolds. In this talk I will explain how this analogy works and why it is useful (and, I think, interesting.) I will also present some recent conjectures about these moduli spaces which arise from this way of thinking.
- Thursday January 31 at 2pm in Ry358
- Benson Farb, University of Chicago
- Teichmuller geometry of moduli space and the Deligne-Mumford compactification
- Abstract: In this talk I will explain some recent joint work with Howard Masur. We relate two important but disparate topics in the study of the moduli space M_g of genus g Riemann surfaces: Teichmuller geometry and the Deligne-Mumford compactification M_g. In particular, we reconstruct M_g purely from the intrinsic metric geometry of M_g.
- Thursday February 7 at 2pm in Ry358
- Anne Thomas, Cornell University
- Existence, covolumes and commensurators of lattices acting on polyhedral complexes
- Abstract: We compare several properties of lattices in semisimple Lie groups and lattices in automorphism groups of polyhedral complexes, such as Davis complexes and right-angled buildings. Questions considered include existence of lattices, their covolumes and (in joint work with A. Barnhill) their commensurators.
- Thursday February 14 at 2pm in Ry358
- Dave Witte Morris, University of Lethbridge
- Noncompact locally symmetric subspaces of noncompact locally symmetric spaces
- Abstract: It has long been known that only two manifolds are minimal in the category of symmetric spaces X = G/K of rank greater than 1. (We assume G is a connected, semisimple Lie group with no compact factors.) Namely, every symmetric space in this category contains either the product of two hyperbolic planes or the symmetric space associated to SL(3,R). The corresponding problem for noncompact spaces of finite volume that are locally symmetric, rather than symmmetric, was solved in joint work with Lucy Lifschitz and Vladimir Chernousov, but infinitely many manifolds are minimal in this category. The compact case remains open.
- Thursday February 21 at 2pm in Ry358
- Azer Akhmedov, UC Santa Barbara
- Perturbations of wreath products and quasi-isometric rigidity
- Abstract: By introducing the notion of perturbation of wreath product, I have succeeded to show that properties like containing free subgroup, free sub-semigroup, satisfying a law, and many other algebraic properties of groups fail to be invariants of quasi-isometry. The constructions are case by case, but they all go along certain general scheme.
  In my talk, I will discuss some key ideas of the construction. Some of the ingredients, like traveling salesman groups, are interesting independently, and turn out to be useful in other areas as well, e.g. in the theory of amenable groups. Also, the properties in question do not imply any (strong) finiteness condition for groups and it seems that one can unite all of the above in one general statement; I will end the talk with some speculations.
- Special day and time
- Monday March 10 at 1:30pm in E202
- Wolfgang Lueck, Universitaet Muenster
- On the Borel Conjecture and related topics
- Abstract: The Borel Conjecture predicts for two closed aspherical manifolds M and N that they are homeomorphic if and only if their fundamental groups agree and that in this case every homotopy equivalence is homotopic to a homeomorphism. This may be viewed as the topological version of Mostow rigidity. It is more or less closely related to the Farrell-Jones Conjecture about the algebraic K- and L-theory of group rings and the Baum Connes Conjecture about the topological K-theory of reduced group C^*-algebras. We present the recent work together with Bartels, where we prove the Farrell-Jones Conjecture and hence the Borel Conjecture in dimension greater or equal to for a large class of groups which includes word-hyperbolic groups and CAT(0)-groups and is closed under directed colimits, taking subgroups and direct products. This implies that these conjectures are true for certain interesting groups (Tarsky monsters, groups with expanders, limit groups) and those exotic aspherical manifolds constructed by Mike Davis.
- Thursday April 3 at 2pm in Ry358
- Daniel Groves, University of Illinois at Chicago
- Homomorphisms to mapping class groups
- Abstract: Let S be an orientable surface of finite type, and Mod(S) its mapping class group. If B is a compact space, and G is its fundamental group then the set of isomorphism classes of S-bundles over B is naturally parametrized by
  X_S(G) = Hom(G,Mod(S))/~
  where the equivalence relation is post-composition by conjugation in Mod(S).
  I'll present some results about the structure of the set X_S(G), for an arbitrary finitely presented group G. Morally, X_S(G) being infinite implies that G admits a splitting as a graph of groups, and the structure of the set X_S(G) can (often) be understood via studying splittings of G.
- Thursday April 24 at 2pm in Ry358
- David Dumas, Brown University
- Thursday May 1 at 2pm in Ry358
- Enrico Leuzinger, Universitaet Karlsruhe
- Thursday May 8 at 2pm in Ry358
- Daniel Allcock, University of Texas at Austin

For questions, contact

Benson Farb, farb at math.uchicago.edu
Alex Eskin, eskin at math.uchicago.edu
Shmuel Weinberger, shmuel at math.uchicago.edu
Nathan Broaddus, broaddus at math.uchicago.edu