Negative Curvature in Fiber Bundles and Counting Problems

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Among the many techniques available for exploring complicated geometric systems, often two of the most fruitful are (1) decomposing related structures into their basic building blocks and (2) sampling from those systems to determine their typical behavior. For example, to understand a high dimensional geometric object, one can study the various ways that object fibers (i.e., can be build out of simpler, lower dimensional pieces which are arranged in a predictable way). Even if a complete understanding of the object may be out of reach, one can attempt to comprehend its properties "on average." In this project, the PI seeks to apply similar principles to the study of geometrically significant groups via their action on naturally associated spaces. This pursuit brings to bear methods from geometry, topology, dynamics, and group theory, and seeks to not only understand these objects in an abstract sense, but to produce tractable models which allow for quantitative understanding.In more detail, the PI will carry out projects that investigate the geometry and topology of hyperbolic bundles as well as study geometrically motivated counting problems in finitely generated groups. In the classical setting of a hyperbolic 3-manifold fibering over the circle, this project builds on a program to study the extent to which the veering triangulation (a certain combinatorial construction) effectively encodes the manifold's geometry and topology. Inspired by the elegant nature of these triangulations, the PI will construct a canonical ideal simplicial complex in the setting of free-by-cyclic groups; a setting where a single topological object controlling the group's algebraic decompositions is currently lacking. Finally, the PI will study the "typical" geometry of fibered manifolds, free-by-cyclic groups, and representations of hyperbolic groups. A common thread throughout these investigations will be to demonstrate the extent to which negative curvature features are persistent and, in the appropriate sense, generic.