Fall 2014
September
12 Alex Lubotzky Hebrew University of Jerusalem
19 Alexei Miasnikov Stevens Institute of Technology
26 Holiday
October
03 Holiday
10 Khalid Bou-Rabee The City College of New York/CUNY
Title: Normal systolic growth of linear groups
Abstract: Normal systolic growth (NS growth) measures the complexity of detecting word metric n-balls of G through its finite quotients. We show that in the class of finitely generated linear groups, NS growth is bounded by an exponential function (note that NS growth is not bounded by any function over the class of all finitely generated groups). We then compute the NS growth for any finitely generated nilpotent group (it is precisely polynomial). This talk covers joint work with Yves Cornulier and Daniel Studenmund.
17 Efim Zelmanov University of California, San Diego, Some open problems in the theory of infinite dimensional algebras inspired by combinatorial group theory
Abstract: The basic problems of Combinatorial Group Theory such as (i) presentation of groups via generators and relations, (ii) The Burnside Problem, (iii) growth functions, had a big influence on the theory's of infinite dimensional algebras. We will discuss these links focusing on open problems.
24 Tatiana Smirnova-Nagnibeda University of Geneva, 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.
31 Martin Bridson University of Oxford, 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 constructioncan 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.
November
07 Tim Riley Cornell University, (Room 5417) 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.
14 Yash Lodha Cornell University
21 Jennifer Taback Bowdoin College
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.