## Topology & Group Theory Seminar Vanderbilt University 2010/2011

Organizer: Mark Sapir

Wednesdays, 4:10pm in SC 1310 (unless otherwise noted)

Wednesday, 1 September 2010

Speaker: Mark Sapir (Vanderbilt University)

Title: On the dimension growth of groups

Abstract: This is a joint work with A. Dranishnikov. We prove that the Thompson group $F$ has exponential dimension growth. We also prove that every solvable finitely generated subgroups of $F$ has polynomial dimension growth while some elementary amenable subgroups of $F$ and some solvable groups of class 3 have dimension growth at least $\exp(\sqrt{n})$.

Wednesday, 15 September 2010

Speaker: John Ratcliffe (Vanderbilt University)

Title: Right-angled Coxeter polytopes, hyperbolic 6-manifolds, and a problem of Siegel.

Abstract: This talk will describe joint work with Brent Everitt (University of York) and Steven Tschantz.

By gluing together the sides of eight copies of an all-right angled hyperbolic 6-dimensional polytope, two orientable hyperbolic 6-manifolds, with Euler characteristic -1, are constructed. They are the first known examples of orientable hyperbolic 6-manifolds having the smallest possible volume. A group theoretical proof will be given for our results.

Wednesday, 29 September 2010

Speaker: Tara Davis (Vanderbilt University)

Title: Subgroup Distortion in Zk wr Z

Abstract: In this talk we discuss the effects of subgroup distortion in the wreath products Zk wr Z. We show that for every m in N, there is a 2-generated subgroup of Zk wr Z having distortion function equivalent to n^m. Moreover, every finitely generated subgroup of Zk wr Z has distortion function bounded above by some polynomial. Finally, we show that unlike other solvable groups every finitely generated abelian subgroup of Zk wr Z is undistorted.

Wednesday, 6 October 2010

Speaker: Michael Brandenbursky (Vanderbilt University)

Title: Quasi-morphisms defined by knot invariants.

Abstract: Quasi-morphisms on a group are real-valued functions which satisfy the homomorphism equation up to a bounded error. They are known to be a helpful tool in the study of the algebraic structure of non-Abelian groups.

I will discuss a construction relating

a) certain knot and link invariants in particular, the ones that come from the knot Floer homology and a Khovanov-type homology,

b) braid groups,

c) the dynamics of area-preserving diffeomorphisms of a two-dimensional disc,

d) quasi-morphisms on the group of all such compactly supported diffeomorphisms of the disc.

Wednesday, 20 October 2010

Speaker: Michael Brandenbursky (Vanderbilt University)

Title: Finite type invariants obtained by counting surfaces

Abstract: A Gauss diagram is a simple, combinatorial way to present a knot. It is known that any Vassiliev invariant may be obtained from a Gauss diagram formula that involves counting (with signs and multiplicities) subdiagrams of certain combinatorial types. These formulas generalize the calculation of a linking number by counting signs of crossings in a link diagram.

Until recently, explicit formulas of this type were known only for few invariants of low degrees. I will present simple formulas for an infinite family of invariants arising from the HOMFLY-PT polynomial. I will also discuss an interesting interpretation of these formulas in terms of counting surfaces of a certain genus and number of boundary components in a Gauss diagram. This is a joint work with M. Polyak.

Wednesday, 27 October

Speaker: Alexander Olshanskii

Title: Space functions of groups

Abstract: I have considered space functions s(n) of finitely presented groups G. (These functions have a natural geometric analog.) To define s(n) we start with a word w of length at most n equal to 1 in G and use relations of G for elementary transformations to obtain the empty word; s(n) bounds from above the tape space (or computer memory) one needs to transform any word of length at most n vanishing in G to the empty word.

One of the main obtained results is the following criterion:

A finitely generated group H has decidable word problem of polynomial space complexity if and only if H is a subgroup of a finitely presented group G with a polynomial space function.

Wednesday, 3 November 2010

Speaker: Ioana Suvaina (Vanderbilt University)

Title: ALE Ricci flat Kahler 4-manifolds

Abstract: In 1989 Kronheimer gave a complete classification of the ALE hyper-Kahler 4-manifolds. In this talk I will extend his classification to ALE Ricci flat Kahler 4-manifolds. I will show that these manifolds are related to orbifolds of type C^2/G, whith G a finite subgroup of U(2), which admit a non-trivial deformation. Time permitting, I will explain how these examples play an important role in the study of the moduli space of constant scalar curvature Kahler metrics.

Wednesday, 17 November 2010

Speaker: Denis Osin (Vanderbilt University)

Title: Approximating the first L^2-betti number of residually finite groups.

Abstract: I will explain how the first L2-Betti number of a finitely generated residually finite group can be estimated in terms of ordinary betti numbers of finite index normal subgroups. Some group theoretic applications will be discussed. This is a joint work with Wolfgang Lueck.

Wednesday, December 1

Speaker: Ilya Kazachkov (Vanderbilt University)

Title: Limiting actions on asymptotic cones and fully residually partially commutative groups.

Abstract: An influential theorem of Rips classifies groups that act freely/stably on real trees and is central in a variety of results in geometric group theory. In my talk I will discuss how one can generalise Rips' theory to groups acting on much larger class of spaces. I will present results about groups acting on asymptotic cones of partially commutative groups, deduce the structure of (fully) residually partially commutative groups and discuss some of the possible applications. This is joint work with Montserrat Casals-Ruiz.

Wednesday, December 8

Speaker: Montse Casals (Vanderbilt University)

Title: Universal completions

Abstract: The algebraic closure of a field K played an important role in classical field theory since it provided a universe for the class of finite algebraic extensions of K or, in other words, for the class of fields which are finitely generated as modules over K. The notion of algebraic closure can be generalised to arbitrary categories as follows. Let C be a category of models (of a fixed language) and let lambda be a cardinal number. We term a model H in C lambda-universal in C if every model in C generated by fewer than lambda generators is embeddable in H and conversely, every subgroup of H generated by fewer than lambda generators belongs to C. In this talk I will show that given an arbitrary model M (of a functional language) the aleph_0-universal model exists for the category of models discriminated by M answering a question raised by Baumslag, Miasnikov and Remeslennikov. We will further discuss the aleph_0-universal group for the class of fully residually free groups and give a model-theoretic description of the Lyndon's free group.

Wednesday, January 19

Speaker: Rémi Coulon (Max-Planck-Institut fur Mathematik, Bonn, Germany)

Title: Outer automorphisms of Burnside groups.

Abstract: The free Burnside group of exponent n, B(r,n), is the quotient of the free group of rank r by the subgroup generated by all n-th powers. This group was introduced in 1902 by W. Burnside who asked wether it has to be finite or not. Since the work of P.S. Novikov and S.I. Adian in the late sixties, one knows that for exponent n large enough the answer is no. In this talk, we are interested in the outer automorphisms of B(r,n). Using a geometrical formulation of the small cancellation theory, we explore the following questions : Which elements of Out(B(r,n)) have infinite order ? Does Out(B(r,n)) contain abelian or non-abelian free subgroups ?

Wednesday, February 9

Speaker: Stefan Wenger (University of Illinois at Chicago)

Title: Lipschitz maps to asymptotic cones and isoperimetric inequalities

Abstract: In this talk I will show how existence (or non-existence) of suitable Lipschitz maps to the asymptotic cone of a finitely presented group G is related to a quadratic (or non-quadratic) Dehn function of G. I will explain how such a relationship is used to prove (1) existence of nilpotent groups with Dehn function which does not grow exactly polynomially, and (2) a sharp version of Gromov's theorem that sub-quadratic implies linear isoperimetric function. If time permits I will talk about higher isoperimetric functions and a similar relationship between Lipschitz maps to asymptotic cones and isoperimetric functions of Euclidean type.

Wednesday, February 16

Speaker: Guoliang Yu (Vanderbilt University)

Title: The Novikov conjecture for algebraic K-theory of group algebras

Abstract: I will give a survey of the Novikov conjecture for algebraic K-theory of group algebras. In particular, I will discuss the case of group algebras over certain ring of operators.

Wednesday, February 23

Speaker: Mikhail Belolipetsky (Durham, UK)

Title: Finiteness theorems for arithmetic hyperbolic reflection groups.

Abstract: I will discuss new effective results towards classification of arithmetic hyperbolic reflection groups and show some new high-dimensional examples of such groups.

Wednesday, March 2

Speaker: Curt Kent (Vanderbilt)

Title: Local homotopy properties of asymptotic cones of groups

Abstract: I will present a necessary condition for an asymptotic cone to have uncountable fundamental group and show that every multiple HNN extension of a free group has an asymptotic cone which is simply connected or which has uncountable fundamental group.

Wednesday, March 16

Speaker: Shmuel Weinberger (University of Chicago)

Title: Lipschitz (and related) constants

Abstract: That balls in a Cayley graph grow at worst exponentially is a reflection of the size of the space of maps from a circle into a compact manifold with bounded Lipschitz constant (as a function of the size of that constant). What about maps from S2? or maps to spheres or between general finite complexes? We will describe some examples and partial results and seeming dead ends in the understanding of distortion. The talk should contain more examples than conjectures, and more conjectures than theorems. (Based on work with Steve Ferry.)

Wednesday, March 23

Speaker: Anton Klyachko (Moscow State University, Russia)

Title: Large and symmetric

Abstract: (joint results with E.I.Khukhro, N.Yu.Makarenko, and Yu.B.Melnikova) If an outer (multilinear) commutator identity holds in a large subgroup of a group, then it holds also in a large characteristic subgroup. Similar assertions are valid for algebras and their ideals and even for some geometric objects. Varying the meaning of the word "large", we obtain many interesting facts. As an application, we give a sharp estimate for the `virtual derived length' of (virtually solvable)-by-(virtually solvable) groups.

Wednesday, March 30

Speaker: Dmitri Burago (Penn State University)

Title: Conjugation-invariant norms on groups of geometric origin.

Abstract: We say that a group is bounded if it has a finite diameter with respect to any bi-invariant metric. Algebraic nature of this property is rather obscure. Apart from trivial cases, it turned out that deciding whether a given group is bounded is a surprisingly difficult task. We will discuss boundedness of various groups of diffeomorphisms, state a number of open problems and talk about some possible applications. The talk is based on a joint work with S. Ivanov and L. Polterovich.

Wednesday, April 6

Speaker: Andreas Thom (University of Leipzig, Germany)

Title: Finite-dimensional approximation properties of groups

Abstract: I want to present various finite-dimensional approximation properties of discrete groups and explain applications towards some longstanding conjectures. On the other side, I will present a group which cannot be approximated by finite subgroups of unitary groups with the metric given by the operator norm. The techniques provide also new information about the interplay between the group structure and the topology of U(n). We will see that word-maps can contract the whole group into any given neighborhood of the neutral element.

Wednesday, April 13

Speaker: Mikhail Volkov (Ural State University, Russia)

Title: Trakhtman's Road Coloring Theorem I

Abstract: We shall present a recent advance in the theory of finite automata: Avraam Trahtman's proof of the so-called Road Coloring Conjecture by Adler, Goodwyn, and Weiss. The conjecture that admits a formulation in terms of recreational mathematics arose in symbolic dynamics and has important implications in coding theory. The proof is elementary in its essence but clever and enjoyable.

Wednesday, April 27
Speaker: Mikhail Volkov (Ural State University, Russia)

Title: Trakhtman's Road Coloring Theorem II

Abstract: The end of a complete proof of Trakhtman's theorem

Wednesday, May 4
Speaker: Yuri Bakhturin (Memorial University of Newfoundland, Canada, and Moscow State University, Russia)

Title: Lie superautomorphisms on associative algebras (joint work with Matej Brear and Spela Spenko)

Abstract: The connection between the associative and the Lie structure of an associative algebra was studied by Herstein and some of his students in the 1950's and 1960's. The classification of Lie isomorphisms was partially achieved at that time but a complete solution was obtained much later. This solution and other interesting results can now be found in a recent monograph "Functional identities" (Frontiers in Mathematics. Birkhuser Verlag, Basel, 2007) by Brear, Chebota and Martindale. In this talk we study Lie superhomomorphisms of associative superalgebras, that is associative algebras graded by the integers mod 2. Lie superhomomorphisms are graded linear maps preserving the Lie superbracket, an operation, defined on homogeneous a, b as ab-ba if at least one of a,b is even and ab+ba, otherwise. The main result we are going to report says that any Lie superautomorphism of a central simple associative superalgebra of dimension different from 2 and 4 is the sum of two maps, the first being either an associative superautomorphism or the negative of a superantiautomorphism, while the second is the linear function vanishing on the superbrackets of the elements of the algebra.