Group Geometry and Mapping Class Groups

Suny At Buffalo, Amherst NY

Investigators

Abstract

The mathematical notion of a group captures a fundamental mechanism whose examples include addition, multiplication, rotations in space, and more. Any group can be described as the symmetries of some object, where the group operation comes from combining symmetries. In this and many other ways, groups are deeply related to geometric and topological spaces, and are spaces in their own right. Geometric group theory grows out of this insight. The research in this project centers on subgroups of mapping class groups, which are infinite groups arising from symmetries of topological surfaces. Their subgroups include all right-angled Artin groups, which are themselves fundamental objects. For example, notice that order never matters in addition, always matters in concatenation (a songbird is not a birdsong), and only sometimes matters in multiplication (no between ordinary numbers, yes between the matrices of linear algebra). Right-angled Artin groups include and interpolate between the first two extremes. Both mapping class groups and right-angled Artin groups are important in geometric group theory, and rich enough that their study has applications to larger families of groups, as well as to low-dimensional manifolds. As fundamental mathematics, work in geometric group theory has potential for practical benefits to society. The work of geometric group theorists, building road maps for groups, has had ramifications to cryptography, which is based on the difficulty of retracing one's steps. In addition, right-angled Artin groups are relevant to any algorithmic task in which order matters between some steps and not between others, a well-documented example being robot motion planning. This project is a geometrically-oriented investigation of three interrelated families of subgroups of mapping class groups: right-angled Artin groups, normal subgroups, and 'convex cocompact' or 'stable' subgroups. The goal of the project is to advance knowledge both specific to mapping class groups and relevant to geometric group theory overall. Among normal subgroups, the project aims to understand the spectrum from free, infinite-rank normal subgroups (whose group of automorphisms is large as possible), to normal subgroups with automorphism group consisting of the mapping class group itself (that is, as small as possible), with right-angled Artin groups appearing as normal subgroups between these two extremes. Objects with automorphism group equal to the mapping class group can be considered geometric models for the mapping class group. This work aims to further elucidate what such geometric models may be. The project also aims to advance the study of convex cocompact subgroups of the mapping class group, and their generalizations to other kinds of groups, including right-angled Artin groups, and more generally, groups acting on CAT(0) spaces. The approaches to be employed rely on group actions on various interesting spaces, including CAT(0) cube complexes, curve complexes of surfaces, projection complexes, and "rotating family" machinery within groups acting on hyperbolic spaces. The latter two are axiomatic constructions, so that results about mapping class subgroups acting on these complexes readily translate to more general settings. 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.