Metric Geometry of Groups and Surfaces

Regents Of The University Of Michigan - Ann Arbor, Ann Arbor MI

Investigators

Abstract

Abstract Award: DMS-0906086 Principal Investigator: Moon Duchin This award is funded under the American Recovery and Reinvestment Act of 2009 (Public Law 111-5). The projects in this proposal are clustered around themes in low-dimensional topology, asymptotic geometry, and the dynamics of group actions. Teichmuller space and the mapping class group are objects for special attention, and tools developed for that setting are often promising for wider study in metric geometry and geometric group theory. The PI proposes to build on recent work on flat metrics, their length spectra, and the associated boundary theory, as well as work on synthetic curvature conditions and their applications to random walks. Specific avenues of future investigation include asymptotics on the space of singular flat surfaces; filling inequalities and higher divergence functions adapted from symmetric spaces to group theory; and the use of intersection patterns of metric half-spaces to study the dynamics of isometries. While quite abstract, ideas in this work can also prove beautifully applicable-- for instance, Thurston's theory of flows on surfaces has direct and strong applications to the study of mixing in fluid dynamics. More immediately, though, this research area addresses foundational questions toward understanding geometric structure of complicated spaces in the large, in the small, and changing over time.