On the geometry and bounded cohomology of racks and quandles (Q6590034)

In this interesting paper, the author introduces 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. More precisely, the author proves that if the group of inner automorphisms of a rack is either an \(S\)-arithmetic Chevalley group of higher rank or a semi-simple Lie group with finite center, then the diameter of each of its connected components is finite. However, it is shown that all connected components of a free product of quandles (satisfying some mild hypothesis) have infinite diameter. The author also introduces a bounded cohomology of racks and quandles with real coefficients, relates them to the above metrics, and proves a vanishing result for racks and quandles with amenable group of inner automorphisms. It turns out that functions that are constant on the connected components of a rack give rise to obvious non-trivial bounded classes. The author shows that if the group of inner automorphisms of a rack is bounded and amenable then this is the entire bounded cohomology.