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Geometric Group Theory Seminar

Upcoming talks:


Embedding spaces into asymptotic cones

Speaker

Kamil Duszenko (Wrocław)

Time

14 May 2009 - 3:00pm

Location

JR 139

Abstract

Past talks:


The Charney-Davis conjecture for certain spheres

Speaker

Andy Frohmader (Cornell)

Time

7 May 2009 - 2:30pm

Location

MA 240

Abstract

The Charney-Davis conjecture is an easy to state linear inequality on the face numbers of simplicial spheres (or more generally, homology spheres) that are also flag complexes, that is, complexes for which ever minimal non-face is a two element set. While the conjecture is reasonably well-known, it is only known to hold in a handful of rather narrow cases, and the proofs are often very complicated. We will discuss some known results and give an easy proof of a relatively broad case.
Some examples of buildings constructed via covering spaces

Speaker

Mike Davis (OSU)

Time

3 Mar 2009 - 3:30pm

Location

MA 240

Abstract

Covering space theory can be used to construct new examples of "nonclassical" buildings. For example, by starting from notion of the direct product of buildings one can define free products, graph products and many other similar constructions.
The geometry of twisted conjugacy classes in Lamplighter groups

Speaker

Jennifer Taback (Bowdoin)

Time

17 Mar 2009 - 3:30pm

Location

MA 240

Abstract

Let G be a group, and ɸ∊Aut(G). We say that g1,g2 ∊G are ɸ-twisted conjugate if there is some h∊G so that hg1ɸ(h)-1=g2. The Reidemeister number R(ɸ) is the cardinality of the set of ɸ-twisted conjugacy classes. We say that a group has property R∞ if every group automorphism has infinitely many twisted conjugacy classes. The study of property R∞is motivated by its connections with fixed point theory and the computation of the Nielsen number of a selfmap of a compact manifold.
I will present a geometric proof based on recent work of Eskin, Fisher and Whyte that the lamplighter group Ln has infinitely many twisted conjugacy classes for any automorphism only when n is divisible by 2 or 3, originally proved by Gonçalves and Wong.
This is joint with with Peter Wong.
Manifold covers of orbifolds

Speaker

Ben McReynolds (U. of Chicago)

Time

3 Feb 2009 - 3:30pm

Location

MA 240

Abstract

I will discuss a generalization of Selberg's lemma for the existence of torsion free subgroups inside a finitely generated linear group. I will also mention a few geometric applications.
The geometry of twisted conjugacy classes in Lamplighter groups

Speaker

Jennifer Taback (Bowdoin)

Time

17 Mar 2009 - 3:30pm

Location

MA 240

Abstract

Let G be a group, and ɸ∊Aut(G). We say that g1,g2 ∊G are ɸ-twisted conjugate if there is some h∊G so that hg1ɸ(h)-1=g2. The Reidieister number R(ɸ) is the cardinality of the set of ɸ-twisted conjugacy classes. We say that a group has property R∞ if every group automorphism has infinitely many twisted conjugacy classes. The study of property R∞is motivated by its connections with fixed point theory and the computation of the Nielsen number of a selfmap of a compact manifold.
I will present a geometric proof based on recent work of Eskin, Fisher and Whyte that the lamplighter group Ln has infinitely many twisted conjugacy classes for any automorphism ɸ only when n is divisible by 2 or 3, originally proved by Gonçalves and Wong.
This is joint with with Peter Wong.
On (some) groups (not) acting on (some) symplectic manifolds

Speaker

Swiat Gal (OSU)

Time

2 Dec 2008 - 3:30pm

Location

MA 240

Abstract

We investigate a cocycle on the group of symplectic diffeomorphisms of a symplecticaly aspherical manifold already known to Arnold, Khesin et al. about ten years ago. The main application is an alternative proof of the Polterovich theorem about the distortion of cyclic subgroups in finitely generated groups of Hamiltonian diffeomorphisms.
This is a joit work with Jarek Kedra.
New examples of finitely presented groups with strong fixed point properties

Speaker

Indira Chatterji (OSU/Université d'Orléans)

Time

25 Nov 2008 - 3:30pm

Location

MA 240

Abstract

Using SL∞Z, we show how to build an explicit example of a finitely presented group that cannot act non-trivially on a CAT(0) space of finite dimension. Examples with even stronger fixed point properties has been recently constructed by Arjantseva et al., using other techniques.
This is joint work with Martin Kassabov.
Fundamental pro-groups and Gromov boundaries of simplicially negatively curved groups

Speaker

Jacerk Świątkowski (Wrocław)

Time

18 Nov 2008 - 2:30pm

Location

MA 240

Abstract

I will describe certain exotic property of Gromov boundaries, called pro-π1-saturation, which holds for a class of word hyperbolic groups called 7-systolic. The class is related to the idea of simplicial negative curvature introduced by T. Januszkiewicz and myself. The property says that every closed subset in the boundary is π1-injective in this boundary, in certain shape-theoretic sense. This excludes 2-disc from being a subset of the boundary. 7-systolic groups are the only known ones with boundary of (topological) dimension above 2 and pro-π1-saturated. Nevertheless, I suspect this property to be generic (in some not yet discovered sense) among high-dimensional Gromov boundaries of groups.
Compactly supported cohomology of buildings

Speaker

Mike Davis (OSU)

Time

4 Nov 2008 - 3:30pm

Location

MA 240

Abstract

In 1976 Borel and Serre calculated the compactly supported cohomology of affine buildings. I will explain an analogous formula for general buildings (where we are interested in nonclassical buildings which are neither affine nor spherical).
This is joint work with Jan Dymara, Tadeusz Januszkiewicz, John Meier and Boris Okun.
Hyperbolic graphs of surface groups

Speaker

Honglin Min (OSU)

Time

23 Oct 2008 - 3:30pm

Location

MA 240

Abstract

We find some conditions which are sufficient for the fundamental groups of the graph of surface groups to be hyperbolic. This is an extension result of Mosher's 'a hyperbolic-by-hyperboli hyperbolic group'. By applying these conditions and other methods, we construct an example of such a group which is distinct from all the hyperbolic groups constructed in that paper of Mosher. This means that this example is not commensurate to any surface-by-free group.
Quasi-conformal geometry and word hyperbolic Coxeter groups

Speaker

Marc Bourdon (Lille, France)

Time

21 Oct 2008 - 3:30pm

Location

MA 240

Abstract

For word hyperbolic Coxeter groups we establish that the combinatorial moduli of their boundaries satisfy a hierarchy phenomenum. As an application we obtain that any word hyperbolic Coxeter group whose boundary is a topological 2-sphere is virtually a Kleinan groups. This is a joint work with B. Kleiner.
Euler Characteristics and growth series for Coxeter groups

Speaker

Boris Okun (U. Wisconsin Milwaukee)

Time

16 Oct 2008 - 3:30pm

Location

CC 358

Abstract

t is an old result of Ruth Charney and Mike Davis that the standard growth series of a manifold Coxeter group satisfies reciprocity equation W(1/q)=(-1)^nW(q). An explanation of this phenomenon was suggested by Jan Dymara, who showed that
  1. W(q)=1/χ_q(W) where χ_q(W) is Euler characteristic computed using q-weighted L2-homology, and
  2. For manifolds Poincare duality switches q and 1/q.
Recently Rick Scott found out that reciprocity holds for right-angled manifold Coxeter groups with a larger set of generators. I will try to describe a weighted L2-theory which explains this.

The flat torus theorem for simplicial nonpositive curvature

Speaker

Tomasz Elsner (OSU)

Time

2 and 9 Oct 2008 - 3:30pm

Location

CC 358 / MA 240

Abstract

Studying possible configurations of flats in a non-hyperbolic systolic complex with a geometric group action gives a lot of information about the complex and the group. Considering not only flats, but also flat minimal surfaces, we obtain a systolic version of the Flat Torus Theorem and systolic analogues of C.Hruska's results on CAT(0)-spaces with isolated flats (for systolic spaces one has a natural modification of the Isolated Flats Property). In particular, a cocompact systolic complex has isolated flats if and only if it contains no triplanes.
A reciprocity formula for right-angled Coxeter groups

Speaker

Ric Scott (Santa Clara U.)

Time

23 May 2008 - 3:30pm

Location

MW 154

Abstract

We discuss a reciprocity formula for the generating function of a RACG with "Eulerian" nerve. The generating set we take is not the standard generating set.
Horofunction bordification of the 1-skeleton of a cubing

Speaker

Dan Guralnik (Vanderbilt University

Time

May 6 2008 - 3:30pm

Location

SM 3082

Abstract

We show that the metric compactification of the 1-skeleton of a cubing X is naturally homeomorphic to the cube-boundary of X, and derive a simplified description of the Roller boundary of X.
This is joint work with Uri Bader.
Peak reduction on right-angled Artin groups

Speaker

Matt Day (U. Chicago)

Time

16 Apr 2008 - 3:30pm

Location

MA 240

Abstract

A right-angled Artin group (RAAG) is a group with a finite resentation whose only relations are commutation relations between generators. Free groups and free abelian groups are examples of RAAGs but there are many other examples. M. Laurence proved in 1995 that the automorphism group Aut(G) of a RAAG G is finitely generated. This talk addresses the following question: if G is a RAAG and g is in G, how can I tell if the stabilizer of g in Aut(G) is finitely generated? If G is a free group, such a stabilizer is always finitely generated, as a corollary to the classical peak-reduction theorem of J.H.C. Whitehead. I will discuss generalizations of Whitehead's theorem to an arbitrary RAAG.
The K-theory of Hamiltonian quotients

Speaker

Megumi Harada

Time

9 Apr 2008 - 3:30pm

Location

MA 240

Abstract

Symplectic geometry lies at the crossroads of many exciting areas of research due to its relationship to geometric representation theory, combinatorics, and algebraic geometry, among others. As often happens in mathematics, the presence of symmetry in these geometric structures -- in this context, a Hamiltonian G-action for G a Lie group -- turns out to be crucial in the computation of topological invariants, such as the Betti numbers, the cohomology ring, or the K-theory, of symplectic manifolds which arise as Hamiltonian quotients. In the first half of the talk, I will give a bird's-eye, motivating overview of this subject. In the second half, I will give a brief survey of my recent work on this topic, which includes a generalization of previous results to the case when the symmetry group is infinite-dimensional, being the loop group LG of a Lie group G.
The zero-divisor conjecture and related notions

Speaker

Igor Mineyev (UIUC)

Time

2 Apr 2008 - 3:30pm

Location

MA 240

Abstract

This will be a brief review of the zero-divisor conjecture due to Kaplansky: the product of two nonzero elements in the group ring is always nonzero. (It is false as I state it here, and I would challenge the audience to find a counterexample before coming to the talk.) Surprisingly, among other things this conjecture is related to questions about the Murray-von Neumann dimension of certain Hilbert modules. The talk will provide more questions than answers.
Diffeomorphism groups of smooth orbifolds and its applications

Speaker

Kojun Abe (Shinshu University)

Time

5 Mar 2008 - 4:30pm

Location

CH 0232

Abstract

Let M be a connected smooth manifold and let D(M) denote the group of diffeomorphims of M which are isotopic to the identity through diffeomorphims with compact support. Hermann and Thurston proved that the group D(M) is perfect. The result is relevant to the foliation theory. In this talk we consider the case where M is a smooth orbifold. We describe the first homology group H1 (D(M)) of D(M) which is defined by the quotient group of D(M) by its commutator subgroup. We apply the result to calculate the first homology of the corresponding automorphism groups of smooth G-manifolds or compact Hausdorff foliations. We can also apply it to the case when a Fuchsian subgroup of SL(2, R) acting on the upper half plane. Then we see that the corresponding first homology of the diffeomorphism group of the orbit space describes the geometric properties around the elliptic singularities and the parabolic singularities of the space.
Isoperimetry of group actions

Speaker

Piotr Nowak (Vanderbilt)

Time

20 Feb 2008 - 4:30pm

Location

CH 0232

Abstract

The notion of isoperimetric profiles is a generalization of isoperimetric dimension, which is a large-scale invariant. In the context of discrete groups isoperimetric profiles were introduced by Vershik, but were well-defined only for amenable groups. The purpose of this talk is a definition of an isoperimetric profile of an action of a finitely generated group on a compact Hausdorff space. We show that these profiles share many properties with original invariants for amenable groups/regularly exhaustible open manifolds. We also define the generalized isoperimetric profile of an amenable group via the action of G on its Stone-Cech compactification.
For this last profile we show that it is a quasi-isometry invariant, explore the relation to growth and asymptotic dimension. We also compute the isoperimetric profile for several classes of groups for which the classical profile was not defined, e.g. hyperbolic groups.
Ultraflats and geometric rank of geodesics

Speaker

Jean-François Lafont (OSU)

Time

30 Jan 2008 - 4:30pm

Location

CH 0232

Abstract

I'll explain how, in the presence of Riemannian non-positive curvature, bi-Lipschitz flats inside asymptotic cones can be used to construct parallel orthogonal Jacobi fields. Applications to geometric problems will be discussed. This was joint work with S. Francaviglia (Univ. Pisa).
On the quasi-isometry invariance of the Scott-Swarup JSJ decomposition for groups

Speaker

Diane Vavrichek (University of Michigan)

Time

23 Jan 2008 - 4:30pm

Location

CH 0232

Abstract

We will show that invariance under quasi-isometry of certain types of vertex groups from Scott and Swarup's JSJ decompositions for groups.
On some combinatorial non-positive curvature

Speaker

Damian Osajda (University of Wrocław)

Time

20 Nov 2007 - 3:30pm

Location

BO 317

Abstract

We will define local conditions (in terms of small balls) on a simplicial complex that imply non-positive-curvature-like properties of it. The class of groups acting geometrically by automorphisms on such complexess contains in particular CAT(0) cubical and systolic groups.
Phan theory and beyond

Speaker

Corneliu Hoffman (BGSU and University of Birmingham)

Time

7 Nov 2007 - 4:30pm

Location

Smith Lab 3082

Abstract

Let G be a finite group of Lie type or a Kac-Moody group acting on the corresponding twin building. Phan theory relates to a certain "unitary" involution of the twin building. One studies an incidence geometry (or simplicial complex) induced by this involution. The results allow one to identify the centralizer in G of the involution as the universal completion of an amalgam of (mostly) unitary groups. This provides an identification method for certain finite simple groups. I will review the known results on Phan-like amalgam presentations for groups of Lie type and Kac-Moody groups, and discuss the newer developments.
Spring 2007