• Bulletin of the Korean Mathematical Society
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• v.59 no.2
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• pp.351-360
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• 2022

We prove that a punctured-torus group of hyperbolic 4-space which keeps an embedded hyperbolic 2-plane invariant has a strictly parabolic commutator. More generally, this rigidity persists for a punctured-surface group.

• Communications of the Korean Mathematical Society
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• v.11 no.4
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• pp.933-936
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• 1996

In this paper, we shall prove that the Kim-Kostrikin group is isomorphic to a hyperbolic orbifold group and so it is infinite.

• Journal of the Korean Mathematical Society
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• v.44 no.3
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• pp.615-626
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• 2007

A pair of pants $\sum(0,\;3)$ is a building block of oriented surfaces. The purpose of this paper is to determine the discrete conditions for the holonomy group $\pi$ of hyperbolic structure of a pair of pants. For this goal, we classify the relations between the locations of principal lines and entries of hyperbolic matrices in $\mathbf{PSL}(2,\;\mathbb{R})$. In the level of the matrix group $\mathbf{SL}(2,\;\mathbb{R})$, we will show that the signs of traces of hyperbolic elements playa very important role to determine the discreteness of holonomy group of a pair of pants.

• Journal of the Korean Mathematical Society
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• v.50 no.2
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• pp.425-444
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• 2013

We construct some hyperbolic hyperelliptic space forms whose fundamental groups are generated by only two or three isometries. Each occurring group is obtained from a supergroup, which is an extended Coxeter group generated by plane re ections and half-turns. Then we describe covering properties and determine the isometry groups of the constructed manifolds. Furthermore, we give an explicit construction of space form of the second smallest volume nonorientable hyperbolic 3-manifold with one cusp.

• East Asian mathematical journal
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• v.36 no.5
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• pp.555-560
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• 2020

We prove a new combination theorem for relatively hyperbolic groups by analyzing diagrams over HNN-extensions of relatively hyperbolic groups.

• Bulletin of the Korean Mathematical Society
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• v.46 no.4
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• pp.713-720
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• 2009

We find all left-invariant minimal unit vector fields and strongly normal unit vector fields on a Lie group which is isometric to the hyperbolic space.

• Journal of the Korean Mathematical Society
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• v.37 no.2
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• pp.297-308
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• 2000

We survey results, obtained in the past three years, on characterizing bounded (and Kobayashi-hyperbolic) Reinhardt domains by their automorphism groups. Specifically, we consider the following two situations: (i) the group is non-compact, and (ii) the dimension of the group is sufficiently large. In addition, we prove two theorems on characterizing general hyperbolic complex manifolds by the dimensions of their automorphism groups.

• Journal of the Korean Mathematical Society
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• v.49 no.2
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• pp.343-356
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• 2012

In this paper, we consider the canonical cusps in complex hyperbolic surfaces. We will classify canonical cusps in complex hyper-bolic surfaces and find correspondence between them and 3-dimensiona nilpotent groups. This paper is a sequel of our paper [6].

• Bulletin of the Korean Mathematical Society
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• v.49 no.5
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• pp.1057-1065
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• 2012

We prove that if $\widehat{\Gamma}$ is a co-contraction of ${\Gamma}$, then the right-angled Coxeter group $C(\widehat{\Gamma})$ embeds into $C({\Gamma})$. Further, we provide a graph ${\Gamma}$ without an induced long cycle while $C({\Gamma})$ does not contain a hyperbolic surface group.

• Journal of the Korean Mathematical Society
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• v.43 no.6
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• pp.1143-1158
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• 2006

We consider the projectivization of Minkowski space with the analytic continuation of the hyperbolic metric and call this an extended hyperbolic space. We can measure the volume of a domain lying across the boundary of the hyperbolic space using an analytic continuation argument. In this paper we show this method can be further generalized to find the volume of a domain with smooth boundary with suitable regularity in dimension 2 and 3. We also discuss that this volume is invariant under the group of hyperbolic isometries and that this regularity condition is sharp.