Ken'ichi Ohshika

<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.

<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>