FPSG '11 Finitely Presented Solvable Groups Conference
Joining the Geometric with the Combinatorial
March 17–18, 2011 • 9am to 5:30pm
The City College of New York
The Center for Algorithms and Interactive Scientific Software (CAISS) of the Division of Science and the Grove School of Engineering of the City College of New York, announces a two-day conference on group theory. The conference will explore the structure and geometry of finitely presented solvable groups.
Gromov's work on finitely generated groups of polynomial growth has led to research exploring the extent to which the asymptotic properties of a finitely generated solvable group determine its algebraic structure.
This conference is designed to join two rather different points of view, the geometric and the combinatorial, to interesting new directions for research.
In his address to the International Congress of Mathematicians in 1983, Gromov talked about groups as geometric objects. This followed on his proof in 1981 that finitely generated groups of polynomial growth contain a nilpotent subgroup of finite index, which has helped to focus attention on the extent to which the asymptotic properties of a finitely generated solvable group have on its algebraic structure.
This has led to a number of results dealing with the quasi-isometries and rigidity of solvable groups. Some of these results have been motivated by ideas coming out of the theory of Lie groups, where the semi-simple ones exhibit a rigidity that is not shared by the solvable ones. This ongoing geometric study of what are perhaps the simplest finitely generated solvable groups has put into sharp relief the very nature of finitely presented solvable groups. Although the work of Bieri and Strebel suggests that the structure of these finitely presented solvable groups is fairly restricted, very little is known in general about them, despite the existence of a number of negative algorithmic results. The special case of finitely presented metabelian groups, whose structure is closely connected to the structure of finitely generated modules over polynomial rings, algebraic geometry and seemingly also to Hilbert's Tenth Problem are themselves a particularly fascinating class of groups. There is an intriguing geometric invariant due to Bieri, Strebel and Neumann which distinguishes the finitely-presented metabelian groups from the finitely-generated, infinitely-related ones.
In view of the current situation, this suggests that a conference designed to join two rather different points of view, the geometric and the combinatorial, would lead to interesting new directions for research, joining these two points of view together. In addition, the possible connections with Hilbert's Tenth Problem provide the possibility of finding applications of recursive function theory to the study of finitely generated metabelian groups which are extremely promising.
The program will consist of 50 minute lectures by the invited speakers of both an expository and research nature. The first day will be mainly devoted to the algebraic and combinatorial side and the second day mainly to the geometric side of the theory. Lots of time will be left for discussions between conference participants.
Download the conference poster here.
Thursday - March 17, 2011
9:00 - 9:30 Breakfast / Registration
9:35 - 9:45 Opening Remarks
9:45 - 10:45 Gilbert Baumslag, The City College of New York/CUNY
Finitely generated metabelian groups
The object of this talk, after recalling some well-known results, is to outline some ideas that seem to fit in with the stated objectives of this conference.
10:45 - 11:15 Break
11:15 - 12:15 Alexei Miasnikov, Stevens Institute of Technology
Tarski type problems in solvable groups
I will discuss recent progress on Tarski type problems in the class of solvable groups. What are groups elementarily equivalent to a given finitely generated solvable group? What are relations between logic and geometry for solvable groups? What are limits in Gromov-Hausdorff topology of a given solvable group? Can one compute Zariski (Krull) dimension of a given solvable group? These are the main questions I would like to address. A new and very natural class of "tame" infinitely presented solvable groups occurs as a bi-product of this research.
12:30 - 1:40 Lunch
1:45 - 2:45 Ralph Strebel, University of Fribourg
Finitely presented soluble groups and the Sigma-invariants
Full abstract here.
2:45 - 3:00 Tea
3:00 - 4:00 John Groves, University of Melbourne
Finitely presented abelian-by nilpotent groups
Full abstract here.
4:15 - 5:15 Olga Kharlampovich, Hunter College/CUNY and McGill University
Algorithmic problems in solvable groups
There are two classes of solvable group varieties where the solvability of the word problem is well known: the varieties of nilpotent groups and the varieties of metabelian groups. All nilpotent and metabelian varieties of groups are finitely based. Every finitely generated nilpotent groups is finitely presented, representable by matrices over $Z$ and residually finite. This implies the solvability of the word problem in nilpotent varieties. The word problem in a nilpotent group is solvable in polynomial time. Finitely generated groups in the variety A^2 are finitely presented in this variety and residually finite, hence have solvable word problem (Ph. Hall). The word problem is solvable in polynomial time because such groups are matrix groups. 3-step solvable groups are very complicated from algorithmic point of view. I am going to talk about unsolvability of the main algorithmic problems for finitely presented 3-solvable groups and about new open questions.
Friday - March 18, 2011
9:00 - 9:40 Breakfast / Registration
9:45 - 10:45 Vitaly Romankov, Omsk University
Algorithmic theory of solvable groups
This will be a brief survey of some algorithmic results, some old, some new, about solvable groups.
10:45 - 11:15 Break
11:15 - 12:15 Walter D. Neumann, Barnard College/Columbia University
Groups as geometric objects
This will be an elementary survey of some of the ideas in and results of considering groups as geometric objects.
12:30 - 1:40 Lunch
1:45 - 2:45 Irine Peng, Indiana University
Quasi-isometries of solvable groups
I will discuss some quasi-isometry rigidity results concerning two types of solvable groups: solvable Lie groups and non-polycyclic nilpotent-by-cyclic groups.
2:45 - 3:00 Tea
3:00 - 4:00 Yves de Cornulier, University of Paris-Sud
On the Dehn function of polycyclic groups
We provide a description of the class of polycyclic groups whose Dehn function is polynomial, showing that others have exponential Dehn function. This class is much larger than the class of virtually nilpotent f.g. groups. This is joint work with R. Tessera.
4:15 - 5:15 Bruce Kleiner, New York University
Asymptotic geometry, finite generation of fundamental groups, and harmonic functions
Let X be a complete Riemannian manifold with nonnegative Ricci curvature, or more generally, a manifold (or simplicial complex) which satisfies certain "polynomial growth" type conditions. We study discrete isometric actions G x X ---> X, and the interplay between the structure of X, finite generation of the group G, and harmonic functions. One of the main motivations is Milnor's conjecture that manifolds with nonnegative Ricci curvature have finitely generated fundamental group. This is joint work with Toby Colding and Burkhard Wilking.
FPSG '11 will be held at Steinman Hall (lobby level) of the Grove School of Engineering at the City College of New York. The City College of New York is located at 137th and Convent Avenue; directions to CCNY can be found on the campus website.
If you wish to participate, please download and complete the registration form found below and return it by February 28, 2011, to email@example.com We especially encourage applications from groups historically under-represented in mathematics.
Some financial support is available principally for graduate students and post-doctoral students. Please indicate on your registration form whether you are interested in receiving financial support.
For questions, you may call CAISS at (212) 650-5167.
Gilbert Baumslag, Stuart Margolis, Gretchen Ostheimer, Vladimir Shpilrain, and Sean Cleary.
This event is sponsored by CUNY Chancellor Matthew Goldstein, CCNY President Dr. Lisa Staiano-Coico, CCNY Dean of Engineering Joseph Barba and a grant from the National Science Foundation. This conference is hosted by the Center for Algorithms and Interactive Scientific Software.