Finitely presented solvable groups at The City College of New York, Fall 2010 conference

Cuny City College, New York NY

Investigators

Abstract

This project is to support a conference on Finitely presented solvable groups at The City College of New York, Fall 2010. 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 has 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. Groups arise in many different areas of mathematics, in physics, in chemistry and in chrystallography among other disciplines. One of the reasons for this is that they can be used as a tool for understanding and defining symmetry. Floor patterns in many churches display a symmetry which can be analyzed and better understood by using group theory and it is this kind of symmetry that groups capture, but in a more abstract way. Groups also display an innate symmetry of their own and in recent years efforts have been made to connect geometry to group theory. This is an important area of current research. The objective of this conference is to join the existing combinatorial and theoretical approach of group theory to the geometric approach. The latter requires a great deal of mathematical machinery. The work that is currently being undertaken involves solvable groups introduced by Evariste Galois (who died in 1832 in a duel aged 21) to determine the nature of the solutions of everyday polynomial equations. The aim of the conference is to make the two aspects of group theory available to graduate students, postdocs and interested professionals.