arXiv subjects

Wonyong Jang

Publications and source records attributed to Wonyong Jang.

5 recordsLinked to original sources

arXiv2026

On non-freeness of groups generated by two parabolic matrices with rational parameters: limit points and the orbit test

For $α\in \mathbb{R}$, let $$G_α:= \left< \begin{bmatrix} 1 & 1 \\ 0 & 1 \end{bmatrix} , \begin{bmatrix} 1 & 0 \\ α& 1 \end{bmatrix} \right> < \mathrm{SL}_2 (\mathbb{R}).$$ K. Kim and the first author established the orbit test, which provides a sufficient condition for $G_α$ not to be a free group of rank 2. In this article, we present two main applications of the orbit test. First, using the corresponding modulo homomorphism, we show that the converse of the orbit test does not hold. In particular, we construct explicit counterexamples, all of which are rational. As another application, we prove that $$ \frac{2 n_3 + 2 n_5 - 1}{n_3 n_4 (2 n_5 - 1)} \quad \quad (n_3, n_4 \neq 0)$$ is a limit point of limit points of non-free rational numbers. Moreover, we prove that $$3 + \frac{3}{2 (9 n - 1)} \quad \text{and} \quad 3 + \frac{9 n + 5}{3 (2 n + 1) (9 n + 4)}$$ are non-free rational numbers which converge to $3$. Their construction relies on the orbit test together with a modified Pell's equation.

math.GR↗

arXiv2026

Co-Hopfianity is not a profinite property

We exhibit two finitely generated residually finite groups $G$ and $H$ with isomorphic profinite completions $\widehat{G} \cong \widehat{H}$, such that $G$ is co-Hopfian while $H$ is not. The construction utilizes Wise's residually finite version of the Rips construction applied to a finitely presented acyclic group $U$ with trivial profinite completion and a strong universality property. A key feature of our approach is the construction of $H$ as a preimage subgroup of $G$ which is conjugate to a proper subgroup of itself. This renders the non-co-Hopfianity of $H$ immediate without requiring a detailed structural analysis of the Rips kernel.

math.GR↗

arXiv2026

Bi-orderability and generalized torsion elements from the perspective of profinite properties

Using fiber products, we construct bi-orderable groups from left-orderable groups. As an application, we show that bi-orderability is not a profinite property, answering a question of Piwek and Wykowski negatively. We also show that the existence of a generalized torsion element is not a profinite property.

math.GR↗

arXiv2024

On the kernel of actions on asymptotic cones

Any finitely generated group $G$ acts on its asymptotic cones in natural ways. The purpose of this paper is to calculate the kernel of such actions. First, we show that when $G$ is acylindrically hyperbolic, the kernel of the natural action on every asymptotic cone coincides with the unique maximal finite normal subgroup $K(G)$ of $G$. Secondly, we use this equivalence to interpret the kernel of the actions on asymptotic cones as the kernel of the actions on many spaces at "infinity". For instance, if $G \curvearrowright M$ is a non-elementary convergence group, then we show that the kernel of actions on the limit set $L(G)$ coincides with the kernel of the action on asymptotic cones. Similar results can also be established for the non-trivial Floyd boundary and the $\mathrm{CAT}(0)$ groups with the visual boundary, contracting boundary, and sublinearly Morse boundary. Additionally, the results are extended to another action on asymptotic cones, called Paulin's construction. In the last section, we calculate the kernel on asymptotic cones for various groups, and as an application, we show that the cardinality of the kernel can determine whether the group admits non-elementary action under some mild assumptions.

math.GR↗

arXiv2021

A sequence of algebraic integer relation numbers which converges to 4

Let $α\in \mathbb{R}$ and let $$A=\begin{bmatrix} 1 & 1 \\ 0 & 1\end{bmatrix} \ \text{and} \ B_α = \begin{bmatrix} 1 & 0 \\ α& 1\end{bmatrix}.$$ The subgroup $G_α$ of $\mathrm{SL}_2(\mathbb{R})$ is a group generated by the matrices $A$ and $B_α$. In this paper, we investigate the property of the group $G_α.$ We construct a generalization of the Farey graph for the subgroup $G_α.$ This graph determines whether the group $G_α$ is a free group of rank $2$. More precisely, the group $G_α$ is a free group of rank $2$ if and only if the graph is tree. In particular, we show that if $1/2$ is a vertex of the graph, then $G_α$ is not a free group of rank $2$. Using this, we construct a sequence of real numbers so that the sequence converges to $4$ and each number has the corresponding group that is not a free group of rank $2$. It turns out that the real numbers are algebraic integers.

math.GT↗