NSF Award
2405061
This project will probe fundamental geometric and dynamical phenomena in several mathematical contexts centered around the notion of symmetry. Mathematically, symmetry (as can appear in molecules and in rigid motions of space) is formalized in the construct of a group, a foundational concept in many branches of mathematics. This project focuses on groups that arise in the study of low-dimensional topology and geometry that concern both the structure of the space itself, such as its curvature, distance and volume, and the structure of their associated parameter spaces, which capture the geometric structures supported on the space, or the configurations of points on the space. For the purposes of this project, the two most important classes of examples are graphs and surfaces, and the groups that that can be obtained by combining these objects in basic ways. This project will identify and study how the inherent structure of these groups reveals important geometric and dynamical phenomena. The research activities in this project will be integrated with graduate and postdoctoral training and the development of structured avenues for undergraduate students to engage in mathematical experimentation and exploration.<br/><br/>Specifically, this project investigates the geometry of groups and the dynamics of free group automorphisms. There are several distinct but interrelated goals: Firstly, the project will develop a new theory of geometric finiteness for subgroups of mapping class groups that is tested against examples and understood from the dual perspectives of the intrinsic geometry of surface group extensions and the extrinsic structure of mapping class groups. This is motivated by the theory of Kleinian groups and builds on the PI's recent work in studying surface group extensions associated to lattice Veech groups. Secondly, the project will study the asymptotic behavior of least pseudo-Anosov dilatations and prove that, up to normalization, these numbers accumulate on only finitely many values. This will be accomplished by carefully analyzing the fibrations of individual hyperbolic 3-manifolds and using the fact that all least dilatation pseudo-Anosovs arise as monodromies of only finitely many 3-manifolds. Thirdly, the project will further develop a new theory of orientability of fully irreducible free group automorphisms and study how this interacts with branched covers, stretch factors in finite covers, and polynomial invariants of free-by-cyclic groups. In particular, it will utilize train track theory to show that each fully irreducible automorphism has a canonical orientation double cover. Fourthly, the project will study the Cannon-Thurston maps that encode the boundary structure of hyperbolic group extensions and prove that these maps are uniformly finite-to-one.<br/><br/>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.
