Researchers Target Unsolved Problems

In Geometric Group Theory

An international team that includes Vanderbilt Mathematics Professors Alexander Olshanskii and Mark Sapir has been awarded a major grant by the National Science Foundation to tackle several unsolved problems in the areas of algebra and geometric group theory.

The “Asymptotic and Probabilistic Methods in Geometric Group Theory” project brings together proven leaders in geometric group theory and related fields of mathematics – including two Fields Medalists – in a Focused Research Group. The problems they will study are of central importance in algebra and geometric group theory, and solutions to them will have an effect on many other areas of mathematics.

The researchers offer a diverse range of specialization that is expected to contribute to the solutions of the problems. “Different people approach problems from different points of view,” says Professor Sapir. “There’s a better chance of solving a problem if you apply a variety of perspectives. We have a broad range of experience within our group, which increases our chances of success.”

Some of the questions that will be studied are:

Is there a finitely presented infinite torsion group?
What are the Dehn functions of residually finite groups?
Is every hyperbolic group risidually finite?
The Virtual Haken Conjecture.

An answer to any of these questions would be considered a major advance in the field.

“This award is the culmination of a lot of previous joint mathematical work here at Vanderbilt,” says Professor Olshanskii. “We’re looking forward to making progress in these very interesting areas.”

The three-year grant has an important educational component, as well. Seven conferences will bring all of the collaborators together and attract international researchers in the fields of algebra and geometric group theory. Funding will also be provided for 14 students and post-doctoral fellows, and for young researchers to attend the conferences.

In addition to Sapir and Olshanskii at Vanderbilt, the collaborators include Rostislav Grigorchuk of Texas A&M University; Alex Lubotzky of the Hebrew University of Jerusalem; Fields Medalist Gregory A. Margulis of Yale University; Gilles Pisier of Texas A&M University; and Fields Medalist Efim Zelmanov of the University of California-San Diego. Michael L. Gromov of the Institut des Hautes Études Scientifiques, who has won both the Wolff Prize and the Kyoto Prize in mathematics, will serve as a consultant on the project.

Date Posted: May 22, 2005