Department of Mathematics

Ken'ichi Ohshika

  (大鹿 健一)

Profile Information

Affiliation
教授, 理学部数学科, 学習院大学
名誉教授, 大阪大学
Degree
(BLANK)(The University of Tokyo)

J-GLOBAL ID
200901012308978709
researchmap Member ID
1000366973

Research Interests

 2

Research Areas

 1

Awards

 2

Papers

 55
  • Ken’ichi Ohshika
    Surveys in Geometry I, 389-415, 2022  Peer-reviewedInvited
  • Sang-hyun Kim, Thomas Koberda, Jaejeong Lee, Ken’ichi Ohshika, Ser Peow Tan, Xinghua Gao
    Mathematische Zeitschrift, 299(3-4) 2103-2130, Dec, 2021  Peer-reviewed
    <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, Oct 12, 2021  Peer-reviewed
    <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, Jul 15, 2021  Peer-reviewedInvited
  • Hirotaka Akiyoshi, Ken’ichi Ohshika, John Parker, Makoto Sakuma, Han Yoshida
    Transactions of the American Mathematical Society, 374 1765-1814, Jan 12, 2021  Peer-reviewed

Misc.

 22

Books and Other Publications

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

Professional Memberships

 2

Research Projects

 49