Group Actions on Hyperbolic Spaces

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The collection of symmetries of an object form an algebraic object called a group. A simple example of a group is that of reflections and rotations of a square, as in the case of a 90 degree rotation, which leaves it unchanged. Studying groups can lead to many interesting questions. One could ask, for instance, how many different groups act on a square? What do such groups have in common? Geometric group theory aims to answer such questions by translating the geometric properties of spaces on which a group acts into algebraic properties of the group. The project will use these techniques to work towards understanding certain classes of groups, all of which act on spaces that have a particular geometric structure, called hyperbolicity. This project also seeks to support and encourage student involvement in mathematics, through support for graduate students, outreach to the local community, and support for a seminar series.In more detail, this projects fits into the broad goal of understanding groups that act on hyperbolic, or negatively curved, spaces. This goal is approached in three distinct ways: first, through understanding all actions of a given group on hyperbolic metric spaces; next, through an in-depth study of two particular actions of big mapping class groups, a class of groups in which there has recently been an explosion of interest; and finally, by seeking to prove a strong stability result for quotients of hierarchically hyperbolic groups, a class of groups which can be completely described by their actions on hyperbolic metric spaces. Parts of this project involve tools from other areas of mathematics, including descriptive set theory and (often non-commutative) ring theory.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.