Topology and Geometry from 3-Manifolds to Free Groups

Many problems throughout mathematics (as well as the other sciences) come down to needing to understand the inner workings of complex geometric or dynamical systems. One fruitful approach is to address the question, “Given the most salient or easily measured features of an object, how can we predict its global shape or long term behavior?” The research in this proposal is directed at addressing this question in the low dimensional setting, focusing on the topology and geometry of three-dimensional spaces with connections to dynamics.

At its core, the investigator will employ two basic techniques. The first is to model a complex system with its basic building blocks. For example, his work with Landry and Minsky shows how dynamical properties of important three-dimension flows are combinatorially encoded by a triangulation of the underlying space.

The second is to use probabilistic techniques that aim to understand the “typical” behavior within a complicated dynamical system. This is the focus of his work with Gekhtman and Tiozzo on properties of random symmetries of negatively curved spaces. In conjunction with this research, the proposal also seeks to support student involvement in related fields within the Philadelphia area through new local seminars and the continued mentoring of graduate students and postdocs.

In greater detail, the proposed research projects divide into three themes. The first is to study the topology and geometry of three-manifolds using a new polynomial invariant of veering triangulations. The polynomial extends McMullen’s Teichmuller polynomial to more general pseudo-Anosov flows, detects faces of the Thurston norm ball, and packages relative growth rates of the associated flow.

The second set of projects seeks to extend these insights to study automorphisms of surface groups and free groups. This includes a project, joint with Dowdall, to build a canonical ideal simplicial complex associated to a hyperbolic free-by-cycle group that encodes the various splitting of the group. The final set of projects concerns dynamics in group theory.

First, the investigator will prove general central limit theorems for the distribution of geometric quantities in finitely generated groups when sampling with respect to the word metric. Second, in collaboration with Gupta, the PI plans to solve a conjecture of Handel–Mosher by demonstrating that a typical free group automorphism and its inverse have distinct stretch factors. These projects will also shed light on dynamical properties of Teichmuller and Outer space.

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.