Connections between conjugation quandles and their underlying groups via residual finiteness and the Hopf property

TL;DR

This paper explores the relationship between conjugation quandles and their underlying groups, demonstrating how properties like residual finiteness and the Hopf property transfer or differ, with specific focus on hyperbolic and Baumslag-Solitar groups.

Contribution

It establishes that Hopfian conjugation quandles imply Hopfian underlying groups, shows the non-converse, and characterizes residual finiteness of conjugation quandles of Baumslag-Solitar groups.

Findings

Hopfian conjugation quandles imply Hopfian groups

Conjugation quandles of hyperbolic groups can be non-Hopfian

Residual finiteness of conjugation quandles of Baumslag-Solitar groups characterized

Abstract

We prove that if a conjugation quandle is Hopfian, then its underlying group is also Hopfian. We also show that the converse does not hold by providing an example. This highlights a distinction between conjugation quandles and their underlying groups. While a recent result shows that every hyperbolic group is Hopfian, conjugation quandles of hyperbolic groups can still be non-Hopfian. Furthermore, we examine conjugation quandles of Baumslag-Solitar groups. We show that these quandles are infinitely generated. Hence, to apply the result that every finitely generated residually finite quandle is Hopfian, it is necessary to work with finitely generated quandles. For this purpose, we employ Dehn quandles as subquandles, which allow us to fully characterize the residual finiteness of conjugation quandles of the Baumslag-Solitar groups.