arXiv subjects

KyeongRo Kim

Publications and source records attributed to KyeongRo Kim.

12 recordsLinked to original sources

arXiv2026

Eclipses on Zippers

Calegari and Loukidou introduced zippers, consisting of a disjoint pair of invariant real trees in the boundary of a closed hyperbolic 3-manifold group $π_1(M)$, which ensure the existence of a universal circle. We study the action of $π_1(M)$ on a minimal zipper and prove a fixed point dichotomy: every nontrivial element either fixes a unique point in each tree or acts freely on both. This answers a question of Calegari and Loukidou. As a consequence, there exists an element with exactly one fixed point in each tree.

math.GT↗

arXiv2025

Singularity Selector: Topological Chirality via Non-Abelian Loops around Exceptional Points

Chirality is more than a geometric curiosity; it governs measurable asymmetries across nature, from enantiomer-selective drugs and left-handed fermions in particle physics to handed charge transport in Weyl semimetals. We extend this universal concept to non-Hermitian systems by defining topological chirality, an invariant that emerges whenever an exceptional-points (EP) pair is present. Built from the non-commutative fundamental group and its braid representation, topological chirality acts as a singularity selector: clockwise EP loops occupy a homotopy class that avoids EPs, whereas counter-clockwise mirrors are equivalent only if they cross the EPs themselves. We confirm this binary rule in an optical microcavity and a non-Hermitian topological band. The same two-sheeted topology governs EP pairs in spin systems, photonic crystals and hybrid light-matter structures, where EP encirclements have already been demonstrated, so the framework transfers without alteration and confirms its experimental viability. Our findings lay the cornerstone for interpreting loop-sensitive observables such as spectral vorticity, the complex Berry phase and the non-Abelian holonomy. Finally, a gluing-of-planes construction extends the invariant to an n-sheeted surface hosting 2m EPs, unifying higher-order EP pairs.

quant-ph↗

arXiv2025

The Euler Class and Flux Homomorphisms under Non-Orientability

For an orientable surface with an area form, there are two invariants of area-preserving dynamics, the flux homomorphism and the Calabi invariant. Tsuboi found a remarkable connection between the Calabi invariant on the closed disk and a topological invariant -- the Euler class. In this paper, we investigate a relationship between the Euler class and the flux homomorphism for non-orientable compact surfaces with one boundary component. Furthermore, we prove the simplicity of the kernel of the flux homomorphisms in this non-orientable setting, which implies the non-existence of invariants analogous to the Calabi invariant.

math.GT↗

arXiv2025

Ping-pong dynamics of hyperbolic-like actions with non-simple points

A hyperbolic-like group is a subgroup of $\operatorname{Homeo}_+(S^1)$ such that every non-trivial element has exactly two fixed points, one attracting and one repelling. We investigate the ping-pong dynamics of hyperbolic-like groups, inspired by a conjecture of Bonatti. We show the existence of a proper ping-pong partition for any pair of non-cyclic point stabilizers. More precisely, our results explicitly provide such a ping-pong partition.

math.GT↗

arXiv2025

Quasimorphisms on the group of density preserving diffeomorphisms of the Möbius band

The existence of quasimorphisms on groups of homeomorphisms of manifolds has been extensively studied under various regularity conditions, such as smooth, volume-preserving, and symplectic. However, in this context, nothing is known about groups of `area'-preserving diffeomorphisms on non-orientable manifolds. In this paper, we initiate the study of groups of density-preserving diffeomorphisms on non-orientable manifolds. Here, the density is a natural concept that generalizes volume without concerning orientability. We show that the group of density-preserving diffeomorphisms on the Möbius band admits countably many unbounded quasimorphisms which are linearly independent. Along the proof, we show that groups of density preserving diffeomorphisms on compact, connected, non-orientable surfaces with non-empty boundary are weakly contractible.

math.GT↗

arXiv2025

Topological Switching via Exceptional Point Pairs in an Optical Microcavity Laser

Conventional mode switching mechanisms, which rely on dynamically encircling exceptional points (EPs) through non-adiabatic transitions (NATs), suffer from intrinsic nonlinear dynamics that hinder precise control and reproducibility in experimental settings. Additionally, these methods exhibit low transmission efficiencies due to path-dependent attenuation, limiting their effectiveness in optical switching and logic gate applications. To overcome these limitations, we propose a novel mode switching approach that leverages a pair of EPs configuration in an optical microcavity, characterized by superradiance and subradiance. This mechanism exploits the topological structure of the Riemann surface to enable robust mode switching control and tunable Q-factor through purely adiabatic encircling. Furthermore, topological protection validated via braid isotopy ensures robustness against noise and parametric perturbations, facilitating a compact, robust, and adaptive non-Hermitian system.

physics.optics↗

arXiv2025

Matsumoto dichotomy on foliated $S^1$-bundles

Given an ergodic harmonic measure on a foliated circle bundle over a closed hyperbolic manifold, Matsumoto constructed a map from the fiber circle to the space of nonempty closed subsets of the boundary sphere of the universal cover of the base manifold. This map is well-defined at almost every point. Also, the map is equivariant under two actions of the fundamental group of the base manifold: the holonomy action on the fiber and the action on the space of closed subsets induced by the boundary sphere action. Matsumoto established a dichotomy for these maps, which corresponds to a dichotomy of ergodic harmonic measures. (Indeed, the Matsumoto dichotomy also concerns ergodic harmonic measures on compact hyperbolic laminations.) The dichotomy says that a Matsumoto map either maps each point to a singleton (Type I) or to the entire sphere (Type II). In this paper, we study actions of closed hyperbolic manifold groups on the circle in the context of the Matsumoto dichotomy. We essentially show that the suspension of any action with a non-discrete image cannot admit a Matsumoto map of type I under the condition of a uniformly bounded number of fixed points. As a consequence, we address a question posed in Matsumoto's paper.

math.GT↗

arXiv2023

Uniform difference between the length spectra of Out(F2) and the genus two handlebody group

In this paper, we analyze the natural homomorphism from the genus g handlebody group to the outer automorphism group of the free group with rank g, in terms of length spectra. In general, the preimage of each fully irreducible outer automorphism contains a potentially infinite number of pseudo-Anosov mapping classes. Our study reveals a crucial relationship: for any pseudo-Anosov map in this preimage, its stretch factor must equal or exceed that of the corresponding fully irreducible outer automorphism. Notably, in the case of genus two, we establish that the minimum stretch factor among these pseudo-Anosov maps is less than ten times the stretch factor of the fully irreducible outer automorphism. These results partially address a question by Hensel and have practical implications, including a lower bound for the geodesic counting problem in the genus two handlebody group.

math.GT↗

arXiv2023

Characterization of Fuchsian groups as laminar groups & the structure theorem of hyperbolic 2-orbifolds

Thurston and Calegari-Dunfield showed that the fundamental group of some tautly foliated hyperbolic 3-manifold acts on the circle in a distinctive way that the action preserves some structure of S^1, so-called a circle lamination. Indeed, a large class of Kleinian groups acts on the circle preserving a circle lamination. In this paper, we are concerned with the converse problem that a group acting on the circle with at least two invariant circle laminations is Kleinian. We prove that a subgroup of the orientation preserving circle homeomorphism group is a Fuchsian group whose quotient orbifold is not a geometric pair of pants (or turnover) if and only if it preserves three circle laminations with a certain transversality. This is the complete generalization of the previous result of the first author which is proven under the assumption that the subgroup is discrete and torsion-free. On the way to our main result, we also show a structure theorem (generalized pants decomposition) for complete 2-dimensional hyperbolic orbifolds, including the case of infinite type, which is of independent interest.

math.GT↗

arXiv2022

Groups acting on veering pairs and Kleinian groups

We show that some laminar group which has an invariant veering pair of laminations is a hyperbolic 3-orbifold group. On the way, we show that from a veering pair of laminations, one can construct a loom space (in the sense of Schleimer-Segerman) as a quotient. Our approach does not assume the existence of any 3-manifold to begin with so this is a geometrization-type result, and supersedes some of the results regarding the relation among veering triangulations, pseudo-Anosov flows, taut foliations in the literature.

math.GT↗

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↗

arXiv2019

Laminar groups and 3-manifolds

Thurston showed that the fundamental group of a close atoroidal 3-manifold admitting a co-oriented taut foliation acts faithfully on the circle by orientation-preserving homeomorphisms. This action on the circle is called a universal circle action due to its rich information. In this article, we first review Thurston's theory of universal circles and follow-up work of other authors. We note that the universal circle action of a 3-manifold group always admits an invariant lamination. A group acting on the circle with an invariant lamination is called a laminar group. In the second half of the paper, we discuss the theory of laminar groups and prove some interesting properties of laminar groups under various conditions.

math.GT↗