In this talk, I will discuss the use of Freudenthal compactifications of infinite-type surfaces and their uncountable fundamental groups to study their mapping class groups. In particular, when S is an orientable infinite type surface S with no planar ends and without boundary, the (extended) mapping class group of S is isomorphic to the mapping class group of the Freudenthal compactification of S. We show that such compactifications are the quotient of a disk by a countable edge-pairing on the boundary circle and thus may be constructed by attaching a single 2-cell to a one-dimensional Peano continuum. This provides an avenue for applying established technology for fundamental groups of one-dimensional Peano continua to big mapping class groups. This work is joint with Jesús Hernández Hernández and Curtis Kent.

Local product structure of a measure has been used as a key ingredient for showing various mixing properties of dynamical systems. One prominent example of this is Ornstein and Weiss's proof of the Bernoulli property for geodesic flows in negative curvature. I will discuss recent work establishing local product structure using a nonuniform Gibbs property for a class of equilibrium states for the geodesic flow for some systems beyond negative curvature. This is joint work with Dave Constantine, Alena Erchenko, Noelle Sawyer, and Grace Work.

Dehn surgery is a classical operation in which one converts one three-manifold to another by first removing a solid torus, and then gluing it back in in a different way. The first operation is called "drilling" and the second "filling". Both of these operations have group-theoretic interpretations in the world of hyperbolic and relatively hyperbolic groups. I will explain those interpretations and applications related to the Cannon conjecture about groups acting on the two-sphere. This talk is based on joint work with Groves, Haïssinsky, Osajda, Sisto, and Walsh.