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An extension of the Maskit slice for 4-dimensional Kleinian groups

2007/07/31 by Yoshiaki Araki, Kentaro Ito · 1 citation
Mathematics · #Computer science #Domain (mathematical analysis) #Extension (predicate logic) #Geometric Analysis and Curvature Flows #Geometric and Algebraic Topology #Geometry #Group (periodic table) #Kleinian group #Mathematical analysis #Mathematics #Mathematics and Applications #Physics #Plane (geometry) #Pure mathematics #Space (punctuation) #Torus #math.GT #msc:30F40 #msc:53A30

paper · pdf · doi:10.1090/s1088-4173-08-00187-2

published in Conformal Geometry and Dynamics of the American Mathematical Society 12(14), 199-226 (Serbian Mathematical Society) · 34 pages, 11 figures. v3: The title is changed and some typo are fixed. To appear in Conform. Geom. dyn. The paper including more clear figures can be downloaded from http://www.math.nagoya-u.ac.jp/~itoken/index.html

arxiv created 2008/10/27 · openalex publication_date 2008/12/15 · arxiv updated 2011/07/04 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05

Abstract

Let<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Gamma"><mml:semantics><mml:mi mathvariant="normal">Γ</mml:mi><mml:annotation encoding="application/x-tex">Γ</mml:annotation></mml:semantics></mml:math></inline-formula>be a<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="3"><mml:semantics><mml:mn>3</mml:mn><mml:annotation encoding="application/x-tex">3</mml:annotation></mml:semantics></mml:math></inline-formula>-dimensional Kleinian punctured torus group with accidental parabolic transformations. The deformation space of<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Gamma"><mml:semantics><mml:mi mathvariant="normal">Γ</mml:mi><mml:annotation encoding="application/x-tex">Γ</mml:annotation></mml:semantics></mml:math></inline-formula>in the group of Möbius transformations on the<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="2"><mml:semantics><mml:mn>2</mml:mn><mml:annotation encoding="application/x-tex">2</mml:annotation></mml:semantics></mml:math></inline-formula>-sphere is well known as the Maskit slice<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper M Subscript 1 comma 1"><mml:semantics><mml:msub><mml:mrow class="MJX-TeXAtom-ORD"><mml:mrow class="MJX-TeXAtom-ORD"><mml:mi class="MJX-tex-caligraphic" mathvariant="script">M</mml:mi></mml:mrow></mml:mrow><mml:mrow class="MJX-TeXAtom-ORD"><mml:mn>1</mml:mn><mml:mo>,</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:annotation encoding="application/x-tex">\mathcal M1,1</mml:annotation></mml:semantics></mml:math></inline-formula>of punctured torus groups. In this paper, we study deformations<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Gamma prime"><mml:semantics><mml:msup><mml:mi mathvariant="normal">Γ</mml:mi><mml:mo>′</mml:mo></mml:msup><mml:annotation encoding="application/x-tex">Γ ’</mml:annotation></mml:semantics></mml:math></inline-formula>of<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Gamma"><mml:semantics><mml:mi mathvariant="normal">Γ</mml:mi><mml:annotation encoding="application/x-tex">Γ</mml:annotation></mml:semantics></mml:math></inline-formula>in the group of Möbius transformations on the<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="3"><mml:semantics><mml:mn>3</mml:mn><mml:annotation encoding="application/x-tex">3</mml:annotation></mml:semantics></mml:math></inline-formula>-sphere such that<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Gamma prime"><mml:semantics><mml:msup><mml:mi mathvariant="normal">Γ</mml:mi><mml:mo>′</mml:mo></mml:msup><mml:annotation encoding="application/x-tex">Γ ’</mml:annotation></mml:semantics></mml:math></inline-formula>does not contain screw parabolic transformations. We will show that the space of the deformations is realized as a domain of<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="3"><mml:semantics><mml:mn>3</mml:mn><mml:annotation encoding="application/x-tex">3</mml:annotation></mml:semantics></mml:math></inline-formula>-space<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper R cubed"><mml:semantics><mml:msup><mml:mrow class="MJX-TeXAtom-ORD"><mml:mi mathvariant="double-struck">R</mml:mi></mml:mrow><mml:mn>3</mml:mn></mml:msup><mml:annotation encoding="application/x-tex">\mathbb R3</mml:annotation></mml:semantics></mml:math></inline-formula>, which contains the Maskit slice<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper M Subscript 1 comma 1"><mml:semantics><mml:msub><mml:mrow class="MJX-TeXAtom-ORD"><mml:mrow class="MJX-TeXAtom-ORD"><mml:mi class="MJX-tex-caligraphic" mathvariant="script">M</mml:mi></mml:mrow></mml:mrow><mml:mrow class="MJX-TeXAtom-ORD"><mml:mn>1</mml:mn><mml:mo>,</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:annotation encoding="application/x-tex">\mathcal M1,1</mml:annotation></mml:semantics></mml:math></inline-formula>as a slice through a plane. Furthermore, we will show that the space also contains the Maskit slice<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper M Subscript 0 comma 4"><mml:semantics><mml:msub><mml:mrow class="MJX-TeXAtom-ORD"><mml:mi class="MJX-tex-caligraphic" mathvariant="script">M</mml:mi></mml:mrow><mml:mrow class="MJX-TeXAtom-ORD"><mml:mn>0</mml:mn><mml:mo>,</mml:mo><mml:mn>4</mml:mn></mml:mrow></mml:msub><mml:annotation encoding="application/x-tex">\mathcal M0,4</mml:annotation></mml:semantics></mml:math></inline-formula>of fourth-punctured sphere groups as a slice through another plane. Some of the other slices of the space will be also studied.

Citations