Asymptotic and algorithmic properties of groups

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DMS-0245600Sapir, Mark V.AbstractTitle: Asymptotic and algorithmic properties of groupsThe PIs propose to study asymptotic and algorithmic properties of groups.The topics include a torsion free non-amenable finitely presented groupwithout 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 everytorsion-free hyperbolic group free-by-finite exponent?), n-dimensionaldiagram groups.Algorithmic properties of groups have been a subject of intensive studysince the beginning of the 20th century (Dehn and Tietze). Novikov, Booneand Higman showed deep connections between logic (especially the theory ofrecursive functions) and group theory. A more recent work by Gromov andothers showed a connection between algorithmic and asymptotic properties(especially isoperimetric and growth functions) of groups and the relatedtopological objects. The PIs found even more intimate connections betweencomplexity of algorithms and asymptotic properties of groups. In particular,they have characterized groups whose word problem is in NP in terms of Dehnfunctions and Higman embeddings, found an NP-complete group, etc.. The PIspropose to further study these connections, and their applications to someoutstanding Burnside-type problems. In particular, they propose to useHigman embeddings to construct finitely presented torsion groups and in thestudy of Dehn functions of metabelian groups. Another direction of theirresearch deals with connections between geometry and structure of groups. Inparticular, they propose to study the class of finite dimensional diagramgroups, 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.