Research in Geometry and Topology

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DMS-0203045Mladen BestvinaThe project encompasses different aspects of geometry, topology, and geometric group theory. The unifying theme is the geometric and topological study of spaces with large symmetry groups. Sometimes it is the group that arises first, for example as a symmetry group of an algebraic object, and then a space is constructed with symmetries reflecting this group. This allows for geometric/topological methods tobe used in algebra. Another aspect of the project involves finding hyperbolic structures on 3-dimensional manifolds. The possibility that all (compact) spaces that locally look like our ordinary 3-dimensional space are in fact geometric objects was first conceived by W.P. Thurston and verified in many cases. Yet another aspect involves Diophantinegeometry over groups. This is about a study of logical statements that hold for a particular class of groups. Surprisingly, as shown by the work of Z. Sela, topology enters this study; for example, surfaces appear naturally when one considers free groups.More specifically, the topics of the project are: Singularities of character varieties and topological dynamics on charactervarieties, Explicit bounds in the compactness theorems for spaces of discrete and faithful representations, Weak hyperbolization conjecture and immersed surface laminations in 3-manifolds, Constructions of Kaehler groups of low cohomological dimension, Diophantine geometry over groups,Finiteness properties of the Torelli group, Rigidity theorems for Coxeter, Artin, and related groups.