Applications of Algebraic Topology to Geometric Group Theory, Parametrized Fixed Point Theory and Dynamics

点击翻译按钮获取中文摘要

Proposal: DMS-9971219PI: Ross GeogheganAbstract: Professor Geoghegan is working at the interface of topology, geometryand group theory. In collaboration with Professor Robert Bieri(Frankfurt, Germany) he is investigating new topological properties ofan arbitrary action by a group G on a "non-positively curved" space M.They have introduced the idea of n-connectedness of the action over Mand have found that this is an open condition on the space of all suchactions. For actions with discrete orbits they have identified theirconditions as equivalent to well-known finiteness properties ofstabilizers of points in M thus proving that these are also openproperties. Previous work of many authors on a restricted casesuggests that this should be a promising line of investigation, bringingtopological methods into algebra in a new way. They have set theirsights on a test case, the Moebius action of SL(2, Z[1/m]) on thehyperbolic plane (for a positive integer m). They conjecture that theamount of connectvity is directly related to the number of primefactors of m, and they propose to prove this among other things. Incollaboration with Professor Andrew Nicas (McMaster, Canada) ProfessorGeoghegan will continue his study of parametrized fixed point theoryand its relationship to non-singular flows. They hope to elucidate thegeometrical meaning of some K-theoretic torsions which have beendefined for such flows. They have already achieved part of thatprogram but more remains to be done.Mathematics involves various styles or forms of thought. Among theseare the geometrical (or visual) and the algebraic (or discrete). Forexample, "discrete group theory" (the formal study of symmetries) hastraditionally been part of abstract algebra. But fruitful advances inour understanding of symmetries have been made in recent years throughgeometrizing this kind of algebra. One way of doing this is throughthe notion of "fundamental group of a topological space". If thetopological space is chosen wisely, much about the algebra of the groupwhich was previously invisible and unknown becomes clear throughapplying the very old and deep field of "algebraic topology" to thegroup being studied, via this space. The study of symmetries is basicin many applications of mathematics, particularly to physics. Thisapproach is being used in the present proposal to better understand theinterplay of group theory with geometry and number theory (in the partwith Bieri) and with flows in dynamics (in the part with Nicas).