Characterizing the Residual Finiteness of Conjugation Quandles on the Baumslag-Solitar Groups

Abstract: This article fully characterize the residual finiteness of conjugation quandles of the Baumslag-Solitar groups. The key difference between groups and their derived conjugation quandles is that many conjugation quandles are infinitely generated even when their underlying groups are finitely generated. Notably, we prove that the conjugation quandles of Baumslag-Solitar groups are always infinitely generated. Moreover, we show that if a conjugation quandle is Hopfian, then its underlying group is also Hopfian. From this, we identity the conditions for non-Hopfian conjugation quandles of Baumslag-Solitar groups.