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Jan Kim

Publications and source records attributed to Jan Kim.

5 recordsLinked to original sources
arXiv2025

A non-Hopfian ascending HNN-extension of a finitely presented Hopfian group

We find a non-Hopfian ascending HNN-extension of a finitely presented Hopfian group by providing an explicit construction. This result addresses an analogous question to the one posed by Sapir and Wise, which asks whether there is a non-residually finite ascending HNN-extension of a finitely presented residually finite group. Such an analogy is motivated by Mal'cev's result that every finitely generated residually finite group is Hopfian.

math.GR↗
arXiv2024

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

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.

math.GT↗
arXiv2023

A recipe for constructing non-Hopfian relatively hyperbolic groups with Hopfian peripheral subgroups

Very recently, Kim and Lee presented an example of a non-Hopfian relatively hyperbolic group with a Hopfian peripheral subgroup, demonstrating a counterexample to Osin's well-known question (Problem 5.5). In this paper, we provide a general construction method using the so-called image extension theorem to generate non-Hopfian relatively hyperbolic groups with Hopfian peripheral subgroups. Additionally, we provide two specific examples using this construction method.

math.GR↗
arXiv2019

Non-residually finite groups hyperbolic relative to residually finite subgroups

Let $m$ and $k$ be integers such that $|m|, \, |k| >1$ and $\gcd (m,k)=1$. We show that all Baumslag-Solitar groups $BS(m,mk)$ are non-residually finite groups hyperbolic relative to residually finite subgroups. By a result of Osin (2007), this implies that there exists a non-residually finite hyperbolic group, thus solving a long-standing open problem of Gromov (1987). We also show that all $BS(m, mk)$ are non-Hopfian groups hyperbolic relative to Hopfian subgroups.

math.GR↗