Geometric Group Theory and the Topology of 3-Manifolds

Regents Of The University Of Michigan - Ann Arbor, Ann Arbor MI

Investigators

Abstract

DMS-0203883 G. Peter Scott Geometric Group Theory and the Topology of 3-Manifolds The proposer plans to continue his joint work with Gadde Swarup on connections between geometric group theory and the topology of 3-manifolds. In the 1970's, the theory of the characteristic submanifold of a compact 3-manifold was completed. Starting in the mid 1980's, it began to be apparent that there were analogues of this topological result in the purely algebraic setting. Several authors have worked on this in the last 15 years, but so far none of the results have been truly analogous to the topological results except in very special cases. Now the proposer and Swarup have produced the first true algebraic analogues of the topological results and they plan to continue their investigations in order to see how far their results can be carried. Their arguments and results have already given new insights into the topology of 3-manifolds and led to new results in geometric group theory. The proposer expects that further work in this area will have a significant impact on geometric group theory. The proposer plans to continue his joint work with Gadde Swarup on connections between group theory and the theory of 3-dimensional manifolds. About 40 years ago, some remarkable, and entirely unexpected, work of Stallings first demonstrated the depth of these connections. More recently, work of several authors continued this theme by showing that groups can be decomposed in a fashion closely analogous to the way in which 3-dimensional manifolds can be decomposed. The proposer and Swarup have given a new approach to this area and they plan to continue their investigations in order to see how far their results can be carried. Their arguments and results have already given new insights into the theory of 3-dimensional manifolds and led to new results in group theory. The proposer expects that further work in this area will have a significant impact on group theory.