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
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
(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.
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.