On the geometry and bounded cohomology of racks and quandles

TL;DR

This paper explores the geometric structure and bounded cohomology of racks and quandles, establishing new metrics and cohomological properties, including vanishing results for certain automorphism groups.

Contribution

It introduces a natural family of metrics on racks, defines bounded cohomology for racks and quandles, and links these concepts, providing new insights into their algebraic and geometric properties.

Findings

Metrics are related to bi-invariant metrics on automorphism groups

Bounded cohomology is defined and studied for racks and quandles

Vanishing results are proved for racks with amenable automorphism groups

Abstract

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 certain vanishing result for racks and quandles with amenable group of inner automorphisms.