Asymptotic and algorithmic properties of groups

Vanderbilt University, Nashville TN

Investigators

Abstract

DMS-0245600 Sapir, Mark V. Abstract Title: Asymptotic and algorithmic properties of groups The PIs propose to study asymptotic and algorithmic properties of groups. The topics include a torsion free non-amenable finitely presented group without free non-abelian subgroups, a finitely presented infinite torsion group, Dehn functions of groups including Dehn functions of residually finite and Metabelian groups, isoperimetric functions of aspherical manifolds, residually finite hyperbolic groups, structure properties of hyperbolic groups (is every torsion-free hyperbolic group free-by-finite exponent?), n-dimensional diagram groups. Algorithmic properties of groups have been a subject of intensive study since the beginning of the 20th century (Dehn and Tietze). Novikov, Boone and Higman showed deep connections between logic (especially the theory of recursive functions) and group theory. A more recent work by Gromov and others showed a connection between algorithmic and asymptotic properties (especially isoperimetric and growth functions) of groups and the related topological objects. The PIs found even more intimate connections between complexity of algorithms and asymptotic properties of groups. In particular, they have characterized groups whose word problem is in NP in terms of Dehn functions and Higman embeddings, found an NP-complete group, etc.. The PIs propose to further study these connections, and their applications to some outstanding Burnside-type problems. In particular, they propose to use Higman embeddings to construct finitely presented torsion groups and in the study of Dehn functions of metabelian groups. Another direction of their research deals with connections between geometry and structure of groups. In particular, they propose to study the class of finite dimensional diagram groups, for dimension greater than two which, they believe, include a large class of groups acting ``nicely" on cubical complexes. Structural properties of hyperbolic groups will also be under investigation.