Abstract: The abstract commensurator of a group G is the group of all isomorphisms between finite index subgroups of G up to a natural equivalence relation. Commensurators of lattices in semisimple Lie groups are well understood using strong rigidity results of Mostow, Prasad, and Margulis. We will describe commensurators of lattices in solvable groups, where strong rigidity fails. If time permits, we will extend these results to lattices in certain groups that are neither solvable nor semisimple.
Title: Free Groups and Measure Preservation
Abstract: We establish new characterizations of primitive elements and free factors in free groups, which are based on the distributions they induce on finite groups. This new characterization is related to structural phenomena in the set of f.g. subgroups of a given free group.
More specifically, for every finite group G, a word w in the free group on k generators induces a word map from G^k to G. We say that w is measure preserving with respect to G if given uniform distribution on G^k, the image of this word map distributes uniformly on G. It is easy to see that primitive words (words which belong to some basis of the free group) are measure preserving w.r.t. all finite groups, and several authors have conjectured that the two properties are, in fact, equivalent. In a joint work with O. Parzanchevski, we prove this conjecture.
Feb 6, 11:00-19:00 Alexei Fest: Joint meeting of NY Applied Algebra Colloquium , Model theory , Algebraic Cryptography and NY Group theory seminars, Grad Center, Room 4102
Quasi-isometry classification of the Baumslag-Gersten groups
I will present a quasi-isometry classification of the Baumslag-Gersten groups. Gilbert Baumslag used a group in this family as an example of a non-cyclic one-relator group all of whose finite quotients are cyclic. Gersten showed that the Dehn functions of groups in this family are bounded by towers of exponentials, and Platonov proved that each Dehn function is actually equal to a tower of exponentials. I will describe how the geometric models of these groups are constructed from different Baumslag-Solitar complexes and how this geometry, combined with earlier rigidity results of Farb and Mosher for the solvable Baumslag-Solitar groups BS(1,n), yields a quasi-isometry classification for this family of groups which is perhaps more rigid than expected. I'll include a brief overview of quasi-isometry classification and rigidity results for other families of groups whose geometric models are constructed from Baumslag-Solitar complexes. This is joint work with Tullia Dymarz.
A new solution to the Van Neumann-Day Problem for finitely presented groups
Taming the hydra: the word problem and extreme integer compression
For a finitely presented group, the Word Problem asks for an algorithm which declares whether or not words on the generators represent the identity. The Dehn function is the time-complexity of a direct attack on the Word Problem by applying the defining relations.
A "hydra phenomenon" gives rise to novel groups with extremely fast growing (Ackermannian) Dehn functions. I will explain why, nevertheless, there are efficient (polynomial time) solutions to the Word Problems of these groups. The main innovation is a means of computing efficiently with compressed forms of enormous integers.
This is joint work with Will Dison and Eduard Einstein.
TITLE: Grothendieck pairs and the Infinite Genus Problem
Abstract: It has been known since the 1970s that there exist pairs of finitely presented (fp), residually finite (rf) groups H,G that are
not isomorphic but have the same finite quotients -- i.e. have the same profinite genus. Gilbert Baumslag produced
early examples, which were nilpotent. It was also proved at that time that in the nilpotent and related settings,
there can be only finitely many fp groups in a given genus. It remained unknown whether, in a more general setting, there might exist genera
containing infinitely many non-isomorphic fp, rf groups.
In 2004 Fritz Grunewald and I constructed the first Grothedieck pairs, i.e. pairs as above but with H<G such that
the inclusion map induces an isomorphism of profinite completions. In this lecture I'll explain how refinements of our construction
can be combined with recent advances in the understanding of finiteness properties for fibre products and classical ideas around Nielsen
equivalence to construct infinite classes of fp, rf groups that all lie in the same (strong) profinite genus.
(the talk will be held at the Science Center)
About subgroups in Grigorchuk’s group
Abstract. I’ll survey some old and new results about the subgroup structure of Grigorchuk’s group of intermediate growth.
Remark: The talk will be held at the Science Center on the 4th floor at GC