Deformation spaces of geometric structures

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

Investigators

Abstract

Prof. Canary proposes to study deformation spaces of geometric structures arising from representations of hyperbolic groups into semi-simple Lie groups. Many of these investigations are motivated by the classical study of the Teichmuller space of hyperbolic (or conformal) structures on a surface. For example, the Hitchin component of the space of (conjugacy classes of) representations of a closed surface group into PSL(n,R) has been shown to have many striking resemblances to Teichmuller space. Prof. Canary, in collaboration with Bridgeman, Labourie and Sambarino, has recently developed a mapping class group invariant metric on each Hitchin component which restricts to the Weil-Petersson metric on the Fuchsian locus. Prof. Canary proposes to investigate the properties of this metric and use it as a tool to understand the geometry of the Hitchin component. More generally, he proposes to study metrics on spaces of Anosov representations of word hyperbolic groups into semi-simple Lie groups. Prof. Canary also proposes to study the topology and geometry of deformation spaces of hyperbolic 3-manifolds. The Ending Lamination Theorem provides a classification of the manifolds in these deformation spaces, but as the invariants in this classification are not continuous, it does not provide a parameterization. Recent work has shown that the topology of these spaces has a rich structure and many intriguing questions remain to be studied. Prof. Canary will study deformation spaces of geometric structures on manifolds. A 2-dimensional manifold is a space which looks locally like the 2-dimensional plane, e.g. the surface of a basketball or a pretzel. A geometric structure gives a way of measuring distances and angles on the manifold. It is natural to then study spaces of geometric structures (or shapes) on a fixed manifold. The classical Teichmuller theory studies all hyperbolic geometric structures on a fixed surface. Teichmuller theory has played a central role in several mathematical fields, e.g. complex analysis and dynamics, as well as in physics, especially in string theory. Similarly, a 3-manifold is a space that looks locally like 3-dimensional Euclidean space. The universe we live in is an example of a 3-dimensional manifold. Professor Canary proposes to study deformation spaces of hyperbolic geometric structures on 3-manifolds. More generally, Prof. Canary will study both the geometry and the topology of deformation spaces of geometric structures arising from representations of groups into semi-simple Lie groups. In addition, Prof. Canary will continue his commitment to undergraduate education, by continuing his involvement with curriculum development in courses using inquiry-based learning techniques, and his work mentoring graduate students and postdoctoral assistant professors.