研究者業績

大鹿 健一

オオシカ ケンイチ  (Ken'ichi Ohshika)

基本情報

所属
学習院大学 理学部数学科 教授
大阪大学 名誉教授
学位
理学博士(東京大学)

J-GLOBAL ID
200901012308978709
researchmap会員ID
1000366973

研究キーワード

 2

経歴

 4

受賞

 2

論文

 55
  • Ken’ichi Ohshika
    Surveys in Geometry I 389-415 2022年  査読有り招待有り
  • Sang-hyun Kim, Thomas Koberda, Jaejeong Lee, Ken’ichi Ohshika, Ser Peow Tan, Xinghua Gao
    Mathematische Zeitschrift 299(3-4) 2103-2130 2021年12月  査読有り
    <title>Abstract</title>Let <inline-formula><alternatives><tex-math>$$\Delta $$</tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>Δ</mml:mi> </mml:math></alternatives></inline-formula> be a hyperbolic triangle with a fixed area <inline-formula><alternatives><tex-math>$$\varphi $$</tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>φ</mml:mi> </mml:math></alternatives></inline-formula>. We prove that for all but countably many <inline-formula><alternatives><tex-math>$$\varphi $$</tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>φ</mml:mi> </mml:math></alternatives></inline-formula>, generic choices of <inline-formula><alternatives><tex-math>$$\Delta $$</tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>Δ</mml:mi> </mml:math></alternatives></inline-formula> have the property that the group generated by the <inline-formula><alternatives><tex-math>$$\pi $$</tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>π</mml:mi> </mml:math></alternatives></inline-formula>-rotations about the midpoints of the sides of the triangle admits no nontrivial relations. By contrast, we show for all <inline-formula><alternatives><tex-math>$$\varphi \in (0,\pi ){\setminus }\mathbb {Q}\pi $$</tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>φ</mml:mi> <mml:mo>∈</mml:mo> <mml:mo>(</mml:mo> <mml:mn>0</mml:mn> <mml:mo>,</mml:mo> <mml:mi>π</mml:mi> <mml:mo>)</mml:mo> <mml:mo>\</mml:mo> <mml:mi>Q</mml:mi> <mml:mi>π</mml:mi> </mml:mrow> </mml:math></alternatives></inline-formula>, a dense set of triangles does afford nontrivial relations, which in the generic case map to hyperbolic translations. To establish this fact, we study the deformation space <inline-formula><alternatives><tex-math>$$\mathfrak {C}_\theta $$</tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>C</mml:mi> <mml:mi>θ</mml:mi> </mml:msub> </mml:math></alternatives></inline-formula> of singular hyperbolic metrics on a torus with a single cone point of angle <inline-formula><alternatives><tex-math>$$\theta =2(\pi -\varphi )$$</tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>θ</mml:mi> <mml:mo>=</mml:mo> <mml:mn>2</mml:mn> <mml:mo>(</mml:mo> <mml:mi>π</mml:mi> <mml:mo>-</mml:mo> <mml:mi>φ</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math></alternatives></inline-formula>, and answer an analogous question for the holonomy map <inline-formula><alternatives><tex-math>$$\rho _\xi $$</tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>ρ</mml:mi> <mml:mi>ξ</mml:mi> </mml:msub> </mml:math></alternatives></inline-formula> of such a hyperbolic structure <inline-formula><alternatives><tex-math>$$\xi $$</tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>ξ</mml:mi> </mml:math></alternatives></inline-formula>. In an appendix by Gao, concrete examples of <inline-formula><alternatives><tex-math>$$\theta $$</tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>θ</mml:mi> </mml:math></alternatives></inline-formula> and <inline-formula><alternatives><tex-math>$$\xi \in \mathfrak {C}_\theta $$</tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>ξ</mml:mi> <mml:mo>∈</mml:mo> <mml:msub> <mml:mi>C</mml:mi> <mml:mi>θ</mml:mi> </mml:msub> </mml:mrow> </mml:math></alternatives></inline-formula> are given where the image of each <inline-formula><alternatives><tex-math>$$\rho _\xi $$</tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>ρ</mml:mi> <mml:mi>ξ</mml:mi> </mml:msub> </mml:math></alternatives></inline-formula> is finitely presented, non-free and torsion-free; in fact, those images will be isomorphic to the fundamental groups of closed hyperbolic 3-manifolds.
  • Ken’ichi Ohshika
    Sugaku Expositions 34(2) 123-139 2021年10月12日  査読有り
    <p>The notion of Kleinian groups was first introduced by Poincaré in the 19th century, and their study from a viewpoint of complex analysis was developed by Ahlfors, Bers, Kra, Maskit and Marden among others. After Thurston’s innovative work, topological study of Kleinian groups was started and flourished. In this survey article, we look back on Thurston’s influential work and his famous 10 open questions, and explain their significance and how they have been solved. In the latter half, we shall look at more recent development of the theory of Kleinian groups.</p>
  • Ken’ichi Ohshika
    Topology and Geometry 477-502 2021年7月15日  査読有り招待有り
  • Hirotaka Akiyoshi, Ken’ichi Ohshika, John Parker, Makoto Sakuma, Han Yoshida
    Transactions of the American Mathematical Society 374 1765-1814 2021年1月12日  査読有り

MISC

 22

書籍等出版物

 2
  • 大鹿 健一
    岩波書店 2008年 (ISBN: 9784000061483)
  • 大鹿 健一
    American Mathematical Society 2002年 (ISBN: 082182080X)

共同研究・競争的資金等の研究課題

 49