Second meeting, Bar Ilan University, 3.12.12, building 216

Free Engel Groups

This is a joint work with Arye Juhasz. Free n-Engel groups are defined by the identical relation [x,y,y,...,y]=1 with y repeating n times. The question whether free n-Engel groups are locally nilpotent for n big enough was posed by Prof. B.I.Plotkin in the early 50-ies. It is an analog of the famous Burnside problem asking whether the free Burnside groups defined by the identical relation x^n=1 are locally finite. The Burside problem was solved in the negative for a big enough odd n in the epoch-making work by Novikov and Adian. The (much harder) case of an even n was dealt with much later by Sergei Ivanov.

Our approach uses both combinatorial methods as the original approach of Novikov and Adian and geometric methods introduced later by Ol'shanshkii. The situation for Engel groups is considerably more complicated than for Burnside groups because of the complicated structure of Engel relations and the unavoidable abundance of commuting elements. There are four main parts in our method:

(1) The theory of generalized small cancellations (in the form published in Israel J. Math.).

(2) The combinatorial structure of Engel relations explored by Juhasz and Collins.

(3) The canonical form of elements in groups with Engel relations.

(4) Analysis of the structure of contiguity diagrams.

Sharply 2-transitive linear groups

This is a joint work with Yair Glasner. A group G is sharply 2-transitive if it admits a faithful permutation representation that is transitive and free on pairs of distinct points. Conjecturally, for all such groups there exists a near-field N (i.e. a skew field that is distributive only from the left) such that G is isomorphic to the semidirect product of the multiplicative and additive groups of N. This is well known in the finite case. We prove this conjecture when G < GL(n,F) is a linear group, excluding characteristic 2 in two separate ways.

Actions on hyperbolic-like spaces and random walks

Given a group action, a WPD element is an element behaving similarly to an infinite order element in a hyperbolic group. Pseudo-anosovs in mapping class groups, rank one elements in CAT(0) groups and iwips in $Out(F_n)$ are examples of WPD elements (for certain actions). I'll show that WPD elements are generic in non-virtually cyclic groups containing at least one such element, meaning that simple random walks end up in a non-WPD element with exponentially decaying probability in the length of the random walk. I'll also discuss other results about random walks in the same setting.

Waves on Trees and Positive Entropy

We discuss Lindenstrauss' measure rigidity proof of Arithmetic Quantum Unique Ergodicity, and recent extensions that also shed light on the role (or lack thereof) of spectral multiplicities in the theory. The proof requires a positive entropy property, which we obtain by studying the dispersion and interferences of "waves" on the (p+1)-regular tree. Most of this work is joint with E. Lindenstrauss.

Monotone expanders

Expander graphs are highly connected graphs that turn out to be useful, e.g., in theoretical computer science. We shall give a brief introduction to expanders and specifically to monotone expanders (these are graphs whose edges are defined by partial monotone maps). Unlike general expanders, natural distributions on monotone graphs do not yield expanders. We shall discuss an explicit construction of such graphs (which also gives the only known proof-of-existence of them).

The proof is based on work with Jean Bourgain, and partly follows previous work of Helfgott and Bourgain-Gamburd.

The meeting will take place in the math Department room, building N 216, Bar Ilan University. You are cordially invited.