CAREER: Algebraic, Analytic, and Dynamical Properties of Group Actions on 1-Manifolds and Related Spaces

A group is a mathematical abstraction of symmetries of a physical object or a theoretical space. Groups are fundamental objects in mathematics that also emerge in various applications such as in computer science and physics. The algebraic notion of a group associates to a set a binary operation, like multiplication, which satisfies a list of axioms. Groups emerge naturally as symmetries of various types of concrete or abstract spaces in mathematics. There is an intricate relationship between the geometric properties of these spaces and the algebraic properties of their groups of symmetries. The PI will continue his investigation of the landscape of infinite groups that emerge as symmetries of the most natural spaces in mathematics, the circle and the real line. The PI will organize two research workshops aimed at graduate students, and two research experiences programs for undergraduates. These shall be aimed at training a diverse body of students to become future leaders in mathematics. These activities will incorporate computational methods into the students' mathematical exploration of the landscape of infinite groups. This project is jointly funded by Topology and the Established Program to Stimulate Competitive Research (EPSCoR). The PI will investigate the relationship between the algebraic structure of left orderable groups and the topological and dynamical properties of their actions on 1-manifolds and the cantor space. One goal is to investigate the class of finitely presented, infinite, simple groups, and exhibit new conceptual phenomena. This involves investigating notions such as uniform simplicity, and whether there is a finitely presented infinite simple group that acts on the real line by homeomorphisms. Finally, the PI will investigate a family of closely interconnected open problems emerging in combinatorial group theory. This includes a systematic study of normal generation in the class of finitely generated perfect groups, the conjectured existence of non-abelian free subgroups in non-indicable finitely generated left orderable groups, and fundamental groups of subcomplexes of aspherical 2-dimensional CW complexes. 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.