Groups in Geometry and Topology

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AbstractAward: DMS-0905802Principal Investigator: Michael Kapovich This proposal is a continuation of Kapovich's research ofprevious years in the areas of geometry, topology and geometricgroup theory. Most subjects of the planned research revolvearound geometry of group actions on various spaces and geometricstructures on manifolds, as well as geometry of buildings. Theresearch planned by Kapovich covers geometry of buildings withapplications to representation theory, in particular, Kapovichintends to continue his study of the geometry of the modulispaces of polygonal linkages in symmetric spaces and buildings inrelation to the algebraic groups. Other topics of researchinclude discrete groups in algebraic groups (in particular, thecoherence problem for arithmetic lattices), real-projective andcomplex-projective structures.Groups appear naturally as symmetries of mathematical andphysical objects, like wall-patterns, minerals, snowflakes and,ultimately, the entire universe. This project studies relationbetween algebraic properties of groups and geometry of spaces forwhich groups appear as symmetries. This two-way relation isbeneficial for both group theory (which is a part of algebra) andgeometry. An example of such relation is provided by the theoryof "buildings." Comparing various ways to navigate in buildingscan be mathematically described in terms of certain inequalities.Such inequalities in turn provide answers to various questionsabout symmetry groups of buildings.