CAREER: Groups Acting on Combinatorial Objects

University Of California-Santa Cruz, Santa Cruz CA

Investigators

Abstract

This award supports research in geometry and topology, and more specifically in the area of geometric group theory, which studies the connection between the geometry of an object, and the properties of the group of its symmetries. The PI will study certain groups of symmetries of combinatorial objects. Examples of such objects include trees, i.e. collections of nodes and edges, each connecting a pair of nodes, which contain no closed loops. Other examples are finite dimensional polyhedral complexes, which can be thought of as shapes built out of polyhedral blocks of arbitrary dimension, such as cubes or tetrahedra. The PI will explore the connection between the geometry and topology of those objects, the dynamics of the action of their symmetry groups, and the algebraic structure of the symmetries. The project also has educational components. The PI will be involved in the initiative “Women in Groups, Geometry, and Dynamics” and will organize workshops facilitating collaborative learning and research experience for early career female researchers in those areas. Secondly, the PI will organize an invited speaker series at her institution, focusing on various careers of mathematicians in business, industry, and government. Finally, she will continue her engagement in other professional activities, through organization of conferences, workshops, seminars, development of a new graduate course, and mentorship of a diverse group of students. The first research goal of the project is devoted to the study of actions of Artin groups on CAT(0) spaces with the goal of establishing a rigidity of such actions. In particular, the PI will focus on the braid group on four strands, and classify its proper actions on CAT(0) cube complexes, as well as classify its proper and cocompact actions on general CAT(0) complexes. In the second part of the project, she will investigate the profinite aspects of complexes of groups, and in particular of Artin groups. In the final part of the project, the PI will study the subgroup structure of groups acting geometrically on a product of trees, with the goal of constructing finitely generated infinitely presented subgroups in such groups, establishing their incoherence. 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.