Seminar "Group theory and topology"

Info about previous semesters is here. This semester we shall have seminars on Wednesday, at 4:35 pm in SC1403 (as usual).

February 5. Mike Mihalik (Vanderbilt) “Centralizers of Parabolic Subgroups of Even Coxeter Groups”

This is a joint talk with Patrick Bahls (University of Illinois). The pair (W,S) is a Coxeter system if W is a group with Coxeter presentation <S:R>, where R consists of the relations (st)^{m(s,t)} where m(s,t)=m(t,s), m(s,t)=1 iff s=t (i.e each generator is order 2) and m(s,t) in {1, 2, 3, … , \infty} (here m(s,t)=\infty, simply means st has infinite order). If all m(s,t) are either 1, even or \infty, then W is an even Coxeter group. For (W,S) even and T\subset S we give a complete description of the centralizer and center of <T>. These centralizers and centers are shown to be even Coxeter groups that are convex in (W,S). We also show that for an arbitrary Coxeter system (W,S) and single generator s in S, the centralizer of s is convex in (W,S).

February 12. Ivan Shestakov (Sao Paulo, Brazil - Novosibirsk,Russia) “The Nagata automorphism is wild”

This is a joint talk with Ualbai U. Umirbaev (Astana, Kazakhstan). It is well-known that the automorphisms of polynomial rings and free associative algebras in two variables are "tame", that is, they admit a decomposition into a product of linear automorphisms and the automorphisms of the type (x,y) à (x,y+f(x)).

However, in the case of three or more variables the similar question was open and known as ``The generation gap problem" or ``The tame generators problem". In 1972 Nagata constructed a certain automorphism $\sigma$ of the polynomial ring in three variables and conjectered that it is non-tame or "wild".

The purpose of the present work is to confirm the Nagata conjecture. Our main result states that the tame automorphisms of the polynomial ring in three variables over a field of characteristic 0 are algorithmically recognizable. In particular, the Nagata automorphism \sigma is wild.

The Nagata automorphism \sigma is defined as follows:

\sigma(x)=x+(x^2-yz)z,
\sigma(y)=y+2(x^2-yz)x+(x^2-yz)^2z,
\sigma(z)=z.

February 19. Yuri Bahturin (Vanderbilt and Newfoundland), “Group Gradings on Matrix Algebras”

We give a complete classification of gradings of matrix algebras by finite abelian groups. A full description of such gradings is given also in the case of arbitrary finite groups. Before this work (done in collaboration with S. Sehgal and M.Zaicev) some classification results were available only for cyclic groups. We also present some results in the case of not necessarily finite groups.

Special meeting:

February 27 (3:00 pm). Kim Ruane (Tufts University), “Biautomatic vs. CAT(0) - is there a winner?”

These two classes of groups share several of the same properties, yet the relationship between the two classes seems somewhat elusive. There are groups that are biautomatic but not CAT(0), but the other direction is still open. Many of the CAT(0) groups one thinks of are known to be biautomatic (or at least automatic), but for reasons that really have nothing to do with the group being CAT(0). M. Elder and I tried to find biautomatic structures on certain CAT(0) groups using a very natural language of words on the group coming from the geometry of the CAT(0) space. This approach took us to a very abrupt dead end as we obtained yet another characterization of word hyperbolicity for groups in terms of regularity of certain geometric languages.

March 11. Colloquium (SC 1424, 3:10 pm) Martin Kassabov (Yale), “Kazhdan Property and Finite Graphs”

In this talk I survey several classical results about Kazhdan Property $T$ and apply them to two combinatorial problems involving finite graphs --- construction of family of expanders and working time of product replacement algorithm in computational group theory.

Kazhdan property $T$ originated from the representation theory of Lie groups. Shortly after its introduction it was used by Margulis to construct an explicit example of a family of expanders. Unfortunately, the expanding constant of this family was unknown, because all proofs that a group has a property $T$ were not quantitative, and the expanding constants of this family of expanders was unknown.

In a resent paper, A. Lubotzky and I. Pak showed that Kazhdan property $T$ of the group $SL_n(Z)$ implies that the working time of the product replacement algorithm on
$k$-generated abelian groups is logarithmic in the size of the groups, but its dependence on $k$ was unknown.

A recent result by Y. Shalom gave an explicit bound of the Kazhdan constant for the group $SL_n(Z)$, which lead to quantitative bounds for the constants in these two combinatorial constructions.

March 12. Akbar Rhemtulla (University of Alberta) “An ascending HNN construction for metabelian groups”

Using ascending HNN construction we embed every finitely generated metabelian residually torsion-free-nilpotent group G in a metabelian residually torsion-free-nilpotent group G' in which every (two sided) total order is central. A total order < on a group H is called central if [x,y]^n < x for all x, y in H, 1<x and all integers n. It is known that every total order on a torsion-free nilpotent group is central. It is also true that if every total order on every subgroup of a finitely generated ordered soluble group G is central, then G is nilpotent. In general, a finitely generated metabelian residually torsion-free-nilpotent group may admit non central orders. An example of such a group is the wreath product of two infinite cyclic groups. The above embedding of G in G' allows precisely the central total orders on G to extend to total orders on G.

April 2. Mike Newmann (Australian National University) “Databases for finite groups”

The advent of the computer era has resulted in the development of programs for studying groups. In the spirit of the motto of Hall and Senior

"Theories change, but the groups remain"

this talk will discuss the provision of electronic collections of groups.

April 9. Dubravko Ivansic (Murray State University), “A link complement in the 4-sphere with hyperbolic structure”

It is well known that many noncompact hyperbolic 3-manifolds are topologically complements of links in the 3-sphere. We extend this phenomenon to dimension 4 by exhibiting an example of a noncompact hyperbolic 4-manifold that is topologically the complement of 5 tori in the 4-sphere. We also exhibit examples of hyperbolic manifolds that are complements of 5n tori in a simply-connected 4-manifold with Euler characteristic 2n. All the examples are based on a construction of Ratcliffe and Tschantz, who produced 1171 noncompact hyperbolic manifolds with Euler characteristic 1. Our examples are finite covers of the Ratcliffe-Tschantz manifold with the biggest symmetry group.

The differentiable 4-dimensional Poincare conjecture asserts that every manifold homeomorphic to the 4-sphere is diffeomorphic to the 4-sphere. One approach to find a counterexample to the conjecture is to try to construct a handle decomposition of the 4-sphere that cannot be reduced to the handle decomposition of the standard differentiable 4-sphere. Based on the hyperbolic complement of 5 tori in the 4-sphere, we derive a complicated handle decomposition of the 4-sphere which can be shown to reduce to the standard one. The complexity of the diagram illustrates potential difficulties of attempting to solve the conjecture in the mentioned way.

Monday April 14 (4:10 pm, room TBA). Special colloquium. The talk will have two parts: 50 min general talk plus break plus 45 minutes seminar talk.

Rostislav Grigorchuk (Texas A&M) “The Ihara zeta function of infinite graphs, the KNS spectral measure and integrable maps”

This is a joint work with A. Zuk (University of Chicago)

We define the Ihara zeta function for Cayley graphs of infinite finitely generated groups.We extend the definition of the Ihara zeta function to infinite graphs which are limits of sequences X_n of finite k-regular graphs such that X_{n+1} covers X_n. We associate to such a graph a measure mu with support in
[-1,1] called the Kesten-von Neumann-Serre spectral measure.We present a few examples of computation of zeta function and a measure mu for Schreier graphs of some fractal groups generated by finite automata.These computations are closely related to the integrability of some 2-dimensional mappings which are also in focus of our considerations.

April 16. Roman Mikhailov (Steklov Institute, Moscow) “Filtrations in group (co)homologies and homological methods in the theory of dimension subgroups”

We consider filtrations in group (co)homologies, connected with (transfinite) lower central series in groups and generalized Passi-Stammbach filtrations. We study some generalization of the dimension conjecture, introduce a concept of a cohomological concordance in a group ring and show its applications. As application we prove that the (\omega+1)-th dimension subgroup for a torsion nilpotent group is trivial. Also we consider localization theory with respect to the considered (co)homological filtrations.

April 23. Mark Sapir (Vanderbilt) “Diagram groups and directed 2-complexes: homotopy and homology”

This is a joint talk with Victor Guba (Vologda, Russia). A diagram group is the fundamental group of the space of positive paths with fixed ends of a directed 2-complex. Unlike the ordinary loop space, that space of paths turns out to be a K(.,1). We show how to compute all homology groups of a diagram group. We construct FP_\infty (diagram) groups with virtually arbitrary rational Poincare series. This shows, in particular, that the class of FP_\infty infinitely dimensional groups is large. We find concrete examples of finitely presented diagram groups containing all countable diagram groups as subgroups. Finally, we show that all diagram groups are totally orderable.