大鹿 健一

<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>

© Springer Nature Switzerland AG 2019. All rights reserved. Poincaré is one of the pioneers in non-Euclidean geometry and topology. In this paper, we shall first review his work on non-Euclidean geometry and topology. Then we shall see how his researches in these fields are reflected in his philosophical work, especially his philosophical position often called "conventionalism".

We introduce an asymmetric distance function, which we call the left Hausdorff distance function, on the space of geodesic laminations on a closed hyperbolic surface of genus at least 2. This distance is an asymmetric version of the Hausdorff distance between compact subsets of a metric space. We prove a rigidity result for the action of the extended mapping class group of the surface on the space of geodesic laminations equipped with the topology induced from this distance. More specifically, we prove that there is a natural homomorphism from the extended mapping class group into the group of bijections of the space of geodesic laminations that preserve left Hausdorff convergence and that this homomorphism is an isomorphism.

We propose a novel fingerprint descriptor, namely Möbius moduli, measuring local deviation of orientation fields (OF) of fingerprints from conformal fields, and we propose a method to robustly measure them, based on tetraquadrilaterals to approximate a conformal modulus locally with one due to a Möbius transformation. Conformal fields arise by the approximation of fingerprint OFs given by zero-pole models, which are determined by the singular points and a rotation. This approximation is very coarse, e.g., for fingerprints with no singular points (arch type), the zero-pole model’s OF has parallel lines. Quadratic differential (QD) models, which are obtained from zero-pole models by adding suitable singularities outside the observation window, approximate real fingerprints much better. For example, for arch type fingerprints, parallel lines along the distal joint change slowly into circular lines around the nail furrow. Still, QD models are not fully realistic because, for example along the central axis of arch type fingerprints, ridge line curvatures usually first increase and then decrease again. It is impossible to model this with QDs, which, due to complex analyticity, also produce conformal fields only. In fact, as one of many applications of the new descriptor, we show, using histograms of curvatures and conformality indices (log of the absolute values of the Möbius moduli), that local deviation from conformality in fingerprints occurs systematically at high curvature which is not reflected by state-of-the-art fingerprint models as are used, for instance, in the well-known synthetic fingerprint generation tool SFinGe and these differences robustly discriminate real prints from SFinGe’s synthetic prints.

© Springer International Publishing AG 2017. In this chapter we survey how the notion of manifold was born, elaborated, and developed, starting from Riemann’s Habilitationsvortrag. We also touch upon the philosophical side of this development.

Let G, H be two Kleinian groups with homeomorphic quotients H-3/G and H-3/H. We assume that G is of divergence type, and consider the Patterson-Sullivan measures of G and H. The measurable rigidity theorem by Sullivan and Tukia says that a measurable and essentially directly measurable equivariant boundary map (K) over cap from the limit set Lambda(G) of G to that of H is either the restriction of a Mobius transformation or totally singular. In this paper, we shall show that such (k) over cap always exists. In fact, we shall construct (k) over cap concretely from the Cannon-Thurston maps of G and H.

We consider a compact orientable hyperbolic 3-manifold with a compressible boundary. Suppose that we are given a sequence of geometrically finite hyperbolic metrics whose conformal boundary structures at infinity diverge to a projective lamination. We prove that if this limit projective lamination is doubly incompressible, then the sequence has compact closure in the deformation space. As a consequence we generalise Thurston's double limit theorem and solve his conjecture on convergence of function groups affirmatively.

© 2015, Springer Science+Business Media Dordrecht. Let (Formula presented.) be a Heegaard splitting of a closed orientable 3-manifold M (or a bridge decomposition of a link exterior). Consider the subgroup (Formula presented.) of the mapping class group of Hj consisting of mapping classes represented by orientation-preserving auto-homeomorphisms of Hj homotopic to the identity, and let Gj be the subgroup of the automorphism group of the curve complex CC(S) obtained as the image of (Formula presented.). Then the group (Formula presented.) generated by G1 and G2 acts on CC(S) with each orbit being contained in a homotopy class in M. In this paper, we study the structure of the group G and examine whether a homotopy class can contain more than one orbit. We also show that the action of G on the projective lamination space of S has a non-empty domain of discontinuity when the Heegaard splitting satisfies R-bounded combinatorics and has high Hempel distance.

Let G be a non-elementary word-hyperbolic group acting as a convergence group on a compact metrizable space Z so that there exists a continuous G-equivariant map i : partial derivative G -> Z, which we call a Cannon-Thurston map. We obtain two characterizations (a dynamical one and a geometric one) of conical limit points in Z in terms of their pre-images under the Cannon-Thurston map i. As an application we prove, under the extra assumption that the action of G on Z has no accidental parabolics, that if the map i is not injective, then there exists a non-conical limit point z is an element of Z with vertical bar i(-1)(z)vertical bar = 1. This result applies to most natural contexts where the Cannon-Thurston map is known to exist, including subgroups of word-hyperbolic groups and Kleinian representations of surface groups. As another application, we prove that if G is a non-elementary torsion-free word-hyperbolic group, then there exists x is an element of partial derivative G such that x is not a "controlled concentration point" for the action of G on partial derivative G.

In this paper, which is sequel to [10], we give a generalisation of the second Klein-Maskit combination theorem, the one dealing with HNN extensions, to higher dimension. We give some examples constructed as an application of the main theorem.

We show that closed 3-manifolds with high Heegaard distance and bounded subsurface Heegaard distance are primitive stable when they are regarded as representations from the free group corresponding to the handlebody. This implies that any point on the boundary of Schottky space can be approximated by primitive stable representations corresponding to closed hyperbolic 3-manifolds. © 2014 Elsevier B.V.

In this paper, we give a complete criterion for a discrete faithful representation rho: F-n -&gt; PSL(2,C) to be primitive stable. This will answer Minsky's conjectures about geometric conditions on H-3/(F-n ) regarding the primitive stability of rho.

We consider a quotient space of the Bers boundary of Teichmuller space, which we call the reduced Bers boundary, by collapsing each quasiconformal deformation space lying there into a point.This boundary turns out to be independent of the basepoint, and the action of the mapping class group extends continuously to this boundary.This is an affirmative answer to Thurston's conjecture.He also conjectured that this boundary is homeomorphic to the unmeasured lamination space by the correspondence coming from ending laminations.This part of the conjecture needs some correction: we show that the quotient topology of the reduced Bers boundary is different form the topology induced from the unmeasured lamination space.Furthermore, we show that every auto-homeomorphism on the reduced Bers boundary comes from a unique extended mapping class.We also give a way to determine the limit in the reduced Bers boundary for a given sequence in Teichmuller space.

We show that every auto-homeomorphism of the unmeasured lamination space of an orientable surface of finite type is induced by a unique extended mapping class unless the surface is a sphere with at most four punctures or a torus with at most two punctures or a closed surface of genus 2.

We shall show that for a given homeomorphism type and a set of end invariants (including the parabolic locus) with necessary topological conditions which a topologically tame Kleinian group with that homeomorphism type must satisfy, there is an algebraic limit of minimally parabolic, geometrically finite Kleinian groups which has exactly that homeomorphism type and the given end invariants. This shows that the Bers-Sullivan-Thurston density conjecture follows from Marden's conjecture proved by Agol and Calegari-Gabai combined with Thurston's uniformisation theorem and the ending lamination conjecture proved by Minsky, partially collaborating with Masur, Brock and Canary.

The Riley slice is the space of marked geometrically finite Kleinian groups generated by two parabolic elements freely, which have domains of discontinuity homeomorphic to a four-times punctured sphere. We are interested in its extension: the space of marked free Kleinian groups generated by two parabolic elements, which we call the extended Riley sllice. In this paper, we shall construct uniform models (in the sense of Minsky) for the Kleinian groups in the extended Riley slice. Our construction consists of two distinct methods, one for groups near the boundary and the other for groups away from the boundary (in other words, near an end of the deformation space). As an application, we shall show that the boundary of the Riley slice is a Jordan curve and that the extended Riley slice coincides with the closure of the Riley slice.

In this paper, we generalise the first Klein-Maskit combination theorem to discrete groups of Möbius transformations in higher dimensions. The application of the main theorem is discussed in the last section.

Consider a geometrically finite Kleinian group G without parabolic or elliptic elements, with its Kleinian manifold M = (H3 ∪ Ωg)/G. Suppose that for each boundary component of M, either a maximal and connected measured lamination in the Masur domain or a marked conformal structure is given. In this setting, we shall prove that there is an algebraic limit Γ of quasiconformal deformations of G such that there is a homeomorphism h from IntM to H3/Γ compatible with the natural isomorphism from G to Γ, the given laminations are unrealisable in H 3/Γ, and the given conformal structures are pushed forward by h to those of H3/Γ, Based on this theorem and its proof, in the subsequent paper, the Bers-Thurston conjecture, saying that every finitely generated Kleinian group is an algebraic limit of quasi-conformal deformations of minimally parabolic geometrically finite group, is proved using recent solutions of Marden's conjecture by Agol, Calegari-Gabai, and the ending lamination conjecture by Minsky collaborating with Brock, Canary and Masur. © 2009 The Mathematical Society of Japan.

Let G be the fundamental group of a compact, orientable 3-manifold M. We show that if each piece of the canonical decomposition of M has a geometric structure (e.g. when G contains ℤ2), then either G has infinite dimensional second bounded cohomology or G is virtually solvable.

Let Γ be a torsion-free geometrically finite Kleinian group. In this paper, we investigate which systems of loxodromic conjugacy classes of Γ can be simultaneously made parabolic in a group on the boundary of the quasi-conformal deformation space of Γ. We shall prove that for this, it is sufficient that the classes of the system are represented by disjoint primitive simple closed curves on the ideal boundary of H3/Γ.

Minsky proved that two Kleinian groups G1 and G2 are quasiconformally conjugate if they are freely indecomposable, the injectivity radii at all points of H3/G1, H3/G2 are bounded below by a positive constant, and there is a homeomorphism h from a topological core of H3/G1 to that of H3/G2 such that h and h-1 map ending laminations to ending laminations. We generalize this theorem to the case when G1 and G2 are topologically tame but may be freely decomposable under the same assumption on the injectivity radii. As an application, we prove that if a Kleinian group is topologically conjugate to another Kleinian group which is topologically tame and not a free group, and both Kleinian groups satisfy the assumption on the injectivity radii as above, then they are quasi-conformally conjugate. ©1998 American Mathematical Society.

We give sufficient conditions for a geometrically finite Kleinian group G acting in the hyperbolic space Hn to have co-Hopf property, i.e., not to contain non-trivial proper subgroups isomorphic to itself. We provide examples of freely indecomposable geometrically finite non-elementary Kleinian groups which are not co-Hopf if our sufficient condition does not hold. We prove that any topologically tame non-elementary Kleinian group in dimension 3 can not be conjugate by an isometry to its proper subgroup. © Elsevier, Paris.

Two Kleinian groups Γ1 and Γ2 are said to be topologically conjugate when there is a homeomorphism / : S2 → S2 such that fΓ1f-1 = Γ2. It is conjectured that if two Kleinian groups Γ1 and Γ2 are topologically conjugate, one is a quasi-conformal deformation of the other. In this paper generalizing Minsky's result, we shall prove that this conjecture is true when Γ1 is finitely generated and freely indecomposable, and the injectivity radii of all points of H3/Γ1 and H3/Γ2 are bounded below by a positive constant. © 1996 American Mathematical Society.

In this paper we deal with codimension-1 measured laminations whose leaves are minimal surfaces in geometric 3-manifolds with either SL2R or H 2 × E structures. We call such measured laminations minimal measured laminations. Our main theorem states that in a geometric 3-manifold with an SL2R-structure every class in Ry containing incompressible measured laminations is represented uniquely by a minimal measured lamination. This implies that every incompressible lamination in such a 3-manifold is equivalent to a unique minimal measured lamination, which is vertical with respect to geometric fibering structure. © 1990 by Pacific Journal of Mathematics.