Seminar "Group theory and topology"

Schedule for the Fall semester, 2002 (Thursdays, 2:40 pm in SC1403):

September 12:

Ashot Minasyan
Vanderbilt

"On products of quasiconvex subgroups in hyperbolic groups"

We introduce a new class of quasiconvex subsets of a hyperbolic group, which is closed under finite unions and intersections. We study the properties of products of quasiconvex subgroups, show that such sets are quasiconvex, their finite intersections have a similar algebraic representation and, thus, are quasiconvex too.

September 19:

Goulnara Arjantseva
Universite de Geneve

Growth of groups and languages

Let G be a group and A a finite set of generators for G.The exponential growth rate of the pair (G,A) is the limit as n goes to infinity of the n-th root of the number of elements in B(n) where B(n) is the set of elements of length less than n in G with respect to the word length.

Let G be a non-elementary word hyperbolic group and A a finite set of generators for G. We show that for any infinite normal subgroup H of G, the exponential growth rate of G/H with respect to the natural image of A is strictly less than the exponential growth rate of G with respect to A. This group property is called the growth tightness and it is related with Hopf property for G and the growth tightness of certain languages over A.

A result on the minimal (over all finite generating sets of a group) exponential growth rate of a non-elementary word hyperbolic group is also given.

Colloquium talk at 4:10 in SC1431

Graphs, subgroups, and group actions

Let G be a group and A a finite set of generators for G. One can associate a directed A-labelled graph Gamma to every subgroup H of G. It turns out that this graph carries the essential information on the structure of H. For a free group G the idea behind this association has been developed by J. Stallings in an algebraic topology terminology. For every finitely generated group G on approach proposed by A.Olshanskii and the author is applied.

In this talk we review some known results where such an approach via
graphs is used. Then using our graph technique, we give a sufficient condition for a finitely generated subgroup of a word hyperbolic group G to be free and quasiconvex. Finally, we generalize this result to groups acting by isometries on a delta-hyperbolic space.

September 26:

Ashot Minasyan
Vanderbilt

"On products of quasiconvex subgroups in hyperbolic groups"

(continuation)

October 3:

Denis Osin
Vanderbilt

Algebraic entropy of elementary amenable groups

Abstract: We prove that every elementary amenable group of zero entropy contains a nilpotent subgroup of finite index, or , equivalently, every elementary amenable group of exponential growth is of uniform exponential growth. On the other hand, we show that 0 is an accumulation point of entropies of elementary amenable groups.

October 10:

Cornelia Drutu
University of Lille-1

Diophantine approximation on manifolds

We use the geometry of arithmetic lattices in semisimple groups to compute the Hausdorff dimension of all sets of (simultaneously) very well approximable vectors on some rational quadrics.

Colloquium talk at 4:10 in SC1431

Quasi-isometry invariants and asymptotic cones

Finitely generated groups G become geometric objects when endowed with the word metric (depending on the generating set): the distance between a and b from G is the length of the shortest word representing a^{-1}b.

If G acts "nicely" on a metric space (X,d) (for instance if X is the universal cover of a compact manifold M, G is the fundamental group of M) then G as a metric space is quasi-isometric to (X,d) (a quasi-isometry is a bilipschitz map up to an additive constant).This justifies the interest in the study of groups up to quasi-isometry.

In this talk we shall discuss some quasi-isometry invariants of groups. One of the tools in this study is the asymptotic cone of a metric space. For a metric space (X,d), an asymptotic cone represents ``an image of the space seen from infinitely far away''. We shall present some relations between geometric properties of asymptotic cones and the behavior of quasi-isometry invariants.

October 17:

Alexey Muranov
Vanderbilt

Colored Diagrams and Method for Constructing
Bounded-Simple and Bounded-Generated Groups

A group G is called m-simple iff for any two non-trivial elements g,h in G, h equals some product of no more than m conjugates of g and g^-1. A group G is called bounded-simple iff it is m-simple for some m. A group G is called bounded-generated iff it is a product of finitely many its cyclic subgroups. The existence of a bounded-simple 2-generated group, containing a free non-cyclic subgroup, and the existence of an infinite simple bounded-generated 2-generated group are proven.

October 24:

Alexey Muranov
Vanderbilt

Continuation of the previous talk

October 25 (Friday): Colloquium talk, 4:10, SC1431

Efim Zelmanov
(Yale, San Diego)

Lie algebras graded by root systems.

I will talk about a classification project that includes classical Lie algebras, the Freudenthal-Tits "magic" square and recently discovered infinite dimensional superconformal algebras.

October 31:

No meeting

November 7:

No meeting (GGG conference)

November 14:

Zoran Sunik
University of Nebraska-Lincoln

Examples of automaton groups

Automaton groups act on rooted regular trees by automorphisms (or equivalently, on tree boundaries by isometries). We will present the definition of automaton groups and give several interesting examples that are rather easy to describe but nevertheless have many remarkable properties.

The examples will include the first Grigorchuk group and its many cousins, as well as the Baumslag-Solitar solvable groups, cyclic extensions of free abelian groups, lamplighter groups, etc.

We then elaborate on the role these examples play in the theory of infinite groups. Topics that will be discussed are growth, period growth, amenability, presentations, word and conjugacy problems, spectra, etc.

November 21

Martin Kassabov
Yale University

Explicit Kazhdan constants for SL_n(Z)

The group SL_n(Z) is a classical example of a group having property T. Usually this is proved using the Kazhdan's theorem about lattices in Lie groups, which does not lead to any explicit Kazhdan constants. The value of the Kazhdan constant of SL_n(Z) with respect to the set of elementary matrices plays an important role in the computational group theory. M.Burger studied the Kazhdan constant for SL_3(Z) . In a recent paper, Y. Shalom obtained a lower bound for Kazhdan constant of SL_n(Z) of the form O(n^{-2}).

In this talk I will show how by generalizing his method, a better bound of the form O(n^{-1/2}) can be obtained.Since there exists an upper bound of the same type, this shows that the asymptotic behavior of the Kazhdan constant is exactly n^{-1/2}. This result can be used to improve the known bounds for the spectral gap of the Cayley graph of SL_n(Z) and product replacement algorithm for n generated abelian groups.

December 5

Mark Sapir
(Vanderbilt University)

Tree-graded spaces, tree products of spaces and asymptotic cones of groups

I will present some results about asymptotic cones of groups proved jointly with Cornelia Drutu, Anna Erschler and Denis Osin. A metric space X is called tree-graded if X is a union of pieces X_i, i from I, such that different pieces have at most one common point, and every simple loop is contained in one piece. We show that for every collection of complete geodesic metric spaces X_i, there exists unique universal tree-graded space where every piece is isometric to one of the X_i. This space can be explicitly constructed, it is called the tree product of X_i. We show that for every sufficiently nice metric space X (for example a manifold), there exists a finitely generated group G whose asymptotic cone is isometric to the tree product of copies of X. We construct a finitely generated group with infinitely many non-isometric asymptotic cones. We also show that the asymptotic cones of relatively hyperbolic groups are tree graded where the pieces are the asymptotic cones of the subgroup.

January 4

Mladen Bestvina
(University of Utah)

Measured laminations and group theory

I will discuss some aspects of the recent work of Zlil Sela on Tarski's conjectures from the point of view of measured laminations on finite complexes. In particular, I will prove the following classical theorem of Merzlyakov and discuss Sela's generalizations:

Suppose a free group F of finite rank (say 3) is embedded in a f.g. group G and every homomorphism from F to a free group of rank 2 extends to G. Then F is a retract of G.