Second meeting, Ben Gurion University, 27.12.11
Seminar room -101, building 58, Dept of mathematics
Hidden symmetries of random walks and superrigidity
Margulis's superrigidity results state that if L is lattice in a simple Lie group G of rank at least two, then linear representations of L extend to G. We will discuss an approach to this and similar results, based on hidden symmetries of random walks on G. This is a joint walk with Uri Bader.
The completion of Outer Space
The group of outer automorphisms of the free group, Out(F_n), acts naturally on several geometric objects. There is a proper isometric action on Outer Space which plays the role of a homogeneous space of a Lie group in our setting. There are also several simplicial complexes with natural Out(F_n) actions. Two important examples are the free factor complex and the free splitting complex. Both complexes were recently shown to be Gromov hyperbolic (the free splitting complex by Handel-Mosher and the free factor complex by Bestvina-Feighn). It is still not yet fully understood how these complexes relate to each other and to Outer Space. I will present a proof that the free splitting complex is homeomorphic to the simplicial part of the metric completion of Outer Space. As an application, I will present a new proof of a theorem of Francaviglia-Martino: The isometry group of Outer Space is Out(F_n) for n \geq 3 and PGL(2,Z) for n=2.
Afternoon session
(sponsored jointly by the BGU departmental colloquium)
Equidistribution from fractals
A real number is normal in base n if its base-n expansion behaves statistically in the same manner as a sequence of digits chosen independently and with uniform probability from {0,...,n-1}. E. Borel first showed that a.e. number is normal, but it is notoriously difficult to determine whether any particular number is normal, and one interesting weakening is to ask when a small set, e.g of Hausdorff dimension <1, contains normal numbers. In the late 1950s, Cassels and W. Schmidt studied this question for Cantor sets defined by restricting digits in one base (e.g. the ":standard" midle-1/3 Cantor set) and there have been many extensions of their work. In this talk I will discuss a new method which uses geometeric and dynamical properties of a set, rather than its analytical properties, to show that there are many normal number in it. A sample result is that the middle-1/3 Cantor set contains numbers whose square is normal in base 10. This is joint work with P. Shmerkin.
A continuum of exponents for the rate of escape of random walks on groups.
For every 3/4 \le \beta< 1 we construct a finitely generated group so that the expected distance of the simple random walk from its starting point after n steps is n^{\beta+o(1)}. This answers a question of Vershik, Naor and Peres. In other examples, the speed exponent can fluctuate between any two values in this interval.
Previous examples were only of exponents of the form 1-1/2^k or 1 , and were based on lamplighter (wreath product) constructions. (Other than the standard beta=1/2 and beta=1 which are simply diffusive and ballistic behaviours known for a wide variety of groups) In this lecture we will describe how a variation of the lamplighter construction, namely the permutational wreath product, can be used to get precise bounds on the rate of escape in terms of return probabilities of the random walk on some Schreier graphs. We will then show how groups of automorphisms of rooted trees, related to automata groups , can then be constructed and analyzed to get the desired rate of escape. This is joint work with Balint Virag of the University of Toronto.
No previous knowledge of automaton groups or wreath products is assumed.
You are cordially invited.
