The project explores the geometry and topology of moduli spaces of hyperbolic manifolds, the algebra of their fundamental groups, and applications to adjacent areas. The central focus is the geometry and algebra of the mapping class group of a closed surface, specifically the geometric and algebraic behavior of subgroups and profinite completions of this group, and spaces upon which they act. Much of the project is viewed through the lens of hyperbolic geometry, whose theorems and techniques cast a seductive profile upon the mapping class group. The core of the project falls into three parts: (1) continuation of an ongoing project with C. Leininger, relevant to Gromov's Coarse Hyperbolization Problem, to understand convex cocompact subgroups of mapping class groups of surfaces; (2) continuation of a program of the PI, J. Brock, and C. Leininger to produce purely pseudo-Anosov surface subgroups of mapping class groups and surface bundles over surfaces with hyperbolic fundamental group; (3) work devoted to completion of M. Boggi's program to establish the congruence subgroup property for mapping class groups of surfaces, building upon prior work of the PI.
A moduli space is a collection of geometric objects that is itself a geometric object. A practical example is the collection of all possible arrangements of cell phone towers on the surface of the Earth. One may use the distances between individual towers to define a distance between two arrangements of towers, and the collection of arrangements of towers becomes a geometric object itself, a "space" of arrangements. Geographical constraints limit the feasible configurations of towers, and understanding the geometry of the space of feasible configurations can have direct bearing on which configurations provide the best network coverage. The project is concerned with moduli spaces of hyperbolic manifolds, of which the space of configurations of cell phone towers on the Earth is a special example. There is an intriguing analogy between moduli spaces of hyperbolic manifolds and the hyperbolic manifolds themselves. In other words, there is a sense in which a collection of hyperbolic manifolds may be roughly considered a hyperbolic manifold itself, creating a sort of information feedback loop reciprocally informing the study of both the hyperbolic manifolds and their moduli spaces. It is this analogy that lies at the heart of the project.
