Isoperimetric inequalities and the large-scale geometry of groups

Texas A&M Research Foundation, College Station TX

Investigators

Abstract

Abstract Nowak The proposed project focuses on a number of problems related to isoperimetric inequalities on discrete groups and metric spaces. Sobolev and Poincare inequalities are classical examples of such inequalities, properties like non-exactness or Kazhdan's property (T) can also be viewed as generalized isoperimetric conditions. The main objective of the project is to broaden the understanding of isoperimetric inequalities with coefficients in a C*-algebra on which the group acts by automorphisms and of their geometric implications. In particular, the problem of finding explicit, natural examples of groups carrying such inequalities will be addressed. This will lead to a better understanding of the geometry which lies behind recent progress towards the Novikov and the Baum-Connes conjectures in which exactness and a-T-menability played major roles. The isoperimetric problem is one of the most fundamental problems in mathematics dating back to antiquity. In its simplest form it amounts to finding the largest area on the plane enclosed by a closed curve of fixed length. This classical case has been studied extensively and has numerous applications in analysis and geometry. It is therefore natural to ask for generalizations in which "area" and "perimeter" are replaced by more flexible notions, allowing to implement ideas from the classical case to a more demanding setting. These problems lie at the intersection of several major branches of mathematics and one of the main goals of this project is to foster interactions between the underlying algebraic, analytic and geometric structures. This award is funded under the American Recovery and Reinvestment Act of 2009 (Public Law 111-5).