Topology and Group Theory Seminar: February 12, 2025
Speaker: Jesse Peterson (Vanderbilt)
Title: Property HaHaHaHaHaHaHaHaHaHaHa… HaHaHaHaHaHaHaHaHa… HaHaHaHaHaHa… HaHaHa…
Abstract: A seminal result of Haagerup from 1979 is that the word length function with respect to a free generating set on a free group is conditionally negative definite. Groups possessing a proper conditionally negative definite function have since been said to have the Haagerup property and comprise a large and well-studied class of groups containing all amenable groups in addition to free groups. The Haagerup property is a particularly nice analytic approximation property since it passes to subgroups, is a local property, and is stable under measure-equivalence and W*-equivalence. One drawback however is that it is not in general closed under extensions. In this talk, I will discuss a weakening of the Haagerup property, which still has many nice analytic properties, but is more flexible when it comes to constructions such as group extensions. In fact, I will discuss a hierarchy of such properties indexed by the countable ordinals. This is joint work with Fabian Salinas.