Length, Stable Commutator Length, and Hyperbolic Geometry

Geodesic length and stable commutator length give geometric and topological notions of complexity for nullhomologous elements of the fundamental group of a hyperbolic manifold. The ratio of these complexity measures is a sort of geometric-topological isoperimetric ratio called the stable isoperimetric ratio. In this talk, I will discuss this ratio and describe how it relates to different aspects of the geometry and topology of hyperbolic manifolds. In particular, I will talk about how to control the size of this ratio and discuss a connection to the spectrum of the Hodge Laplacian acting on coexact 1-forms.