On the geometry and bounded cohomology of racks and quandles

In this paper, we introduce and investigate a natural family of metrics on connected components of a rack. The metrics are closely related to certain bi-invariant metrics on the group of inner automorphisms of the rack. We also introduce a bounded cohomology of racks and quandles, relate them to the above metrics and prove a vanishing result for racks and quandles with amenable group of inner automorphisms.