Group Actions on Trees and Boundaries of Trees

University Of Utah, Salt Lake City UT

Investigators

Abstract

Geometric group theory connects two foundational fields of mathematics, namely group theory and geometry. A group can be thought of as the set of symmetries of an object such as a water molecule or a Rubik's Cube. A single group can represent the symmetries of many geometric or topological spaces. If the chosen space is nice enough and can be sufficiently well understood, the characteristics of the spaces can reveal properties inherent to the group. One can take the opposite approach as well. Often one can detect properties of a topological or geometric space by studying groups which describe their symmetries. This project aims to explore these connections between groups and the spaces on which they act. Broader impacts of the project include continued mentoring through various programs and networks, public outreach for middle and high school students, and dissemination of knowledge through conference organization. The focus of this project is primarily on groups acting on infinite trees and on boundaries of infinite trees. This includes large classes of groups; for instance it contains all residually finite groups but also many of the known examples of infinite simple groups. The first goal of the project is to extend the PI's past work to better understand the universe of infinite simple groups. The last 10 years have seen an influx of new and surprising theorems illuminating the understanding of the variety of groups in this class. The PI will study groups in the extended Thompson family using their partial actions on a rooted tree, their full action on the Stein-Farley complex, and their embeddings into the homeomorphism group of the Cantor space. The second goal of the project is to better understand branch and automata groups. Over the last forty years, this class of groups has served as a rich source of exotic yet tractable groups. The PI will use automata theory to investigate questions of growth and torsion and apply the long developed theory of branch groups to work towards a general theory of maximal subgroups of branch groups. The final focus of the project is on group properties coming from topological constructions. The PI will develop a geometric group theory analog of the topological theory of branch coverings as well as explore connections between the general theory of homological stability and topological finiteness properties, first through certain natural subgroups of some big mapping class groups by exploiting actions on highly connected complexes related to the curve complex. 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.