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The realization problem for Jørgensen numbers

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Zeitschriftentitel: Conformal Geometry and Dynamics of the American Mathematical Society
Personen und Körperschaften: Yamashita, Yasushi, Yamazaki, Ryosuke
In: Conformal Geometry and Dynamics of the American Mathematical Society, 23, 2019, 2, S. 17-31
Format: E-Article
Sprache: Englisch
veröffentlicht:
American Mathematical Society (AMS)
Schlagwörter:
Geometry and Topology
Details
Zusammenfassung: <p>Let <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G"> <mml:semantics> <mml:mi>G</mml:mi> <mml:annotation encoding="application/x-tex">G</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be a two-generator subgroup of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper P normal upper S normal upper L left-parenthesis 2 comma double-struck upper C right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="normal">P</mml:mi> <mml:mi mathvariant="normal">S</mml:mi> <mml:mi mathvariant="normal">L</mml:mi> </mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mn>2</mml:mn> <mml:mo>,</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">C</mml:mi> </mml:mrow> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathrm {PSL}(2, \mathbb {C})</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. The Jørgensen number <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper J left-parenthesis upper G right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>J</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>G</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">J(G)</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="upper G"> <mml:semantics> <mml:mi>G</mml:mi> <mml:annotation encoding="application/x-tex">G</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is defined by <disp-formula content-type="math/mathml"> \[ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper J left-parenthesis upper G right-parenthesis equals inf left-brace StartAbsoluteValue normal t normal r squared upper A minus 4 EndAbsoluteValue plus StartAbsoluteValue normal t normal r left-bracket upper A comma upper B right-bracket minus 2 EndAbsoluteValue semicolon upper G equals mathematical left-angle upper A comma upper B mathematical right-angle right-brace period"> <mml:semantics> <mml:mrow> <mml:mi>J</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>G</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo>=</mml:mo> <mml:mo movablelimits="true" form="prefix">inf</mml:mo> <mml:mo fence="false" stretchy="false">{</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo stretchy="false">|</mml:mo> </mml:mrow> <mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="normal">t</mml:mi> <mml:mi mathvariant="normal">r</mml:mi> </mml:mrow> <mml:mn>2</mml:mn> </mml:msup> <mml:mi>A</mml:mi> <mml:mo>−<!-- − --></mml:mo> <mml:mn>4</mml:mn> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo stretchy="false">|</mml:mo> </mml:mrow> <mml:mo>+</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo stretchy="false">|</mml:mo> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="normal">t</mml:mi> <mml:mi mathvariant="normal">r</mml:mi> </mml:mrow> <mml:mo stretchy="false">[</mml:mo> <mml:mi>A</mml:mi> <mml:mo>,</mml:mo> <mml:mi>B</mml:mi> <mml:mo stretchy="false">]</mml:mo> <mml:mo>−<!-- − --></mml:mo> <mml:mn>2</mml:mn> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo stretchy="false">|</mml:mo> </mml:mrow> <mml:mspace width="mediummathspace" /> <mml:mo>;</mml:mo> <mml:mspace width="mediummathspace" /> <mml:mi>G</mml:mi> <mml:mo>=</mml:mo> <mml:mo fence="false" stretchy="false">⟨<!-- ⟨ --></mml:mo> <mml:mi>A</mml:mi> <mml:mo>,</mml:mo> <mml:mi>B</mml:mi> <mml:mo fence="false" stretchy="false">⟩<!-- ⟩ --></mml:mo> <mml:mo fence="false" stretchy="false">}</mml:mo> <mml:mo>.</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">J(G) = \inf \{ |\mathrm {tr}^2 A-4| + |\mathrm {tr} [A,B]-2| \: ; \: G=\langle A, B\rangle \}.</mml:annotation> </mml:semantics> </mml:math> \] </disp-formula> If <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G"> <mml:semantics> <mml:mi>G</mml:mi> <mml:annotation encoding="application/x-tex">G</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is a non-elementary Kleinian group, then <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper J left-parenthesis upper G right-parenthesis greater-than-or-equal-to 1"> <mml:semantics> <mml:mrow> <mml:mi>J</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>G</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo>≥<!-- ≥ --></mml:mo> <mml:mn>1</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">J(G)\geq 1</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. This inequality is called Jørgensen’s inequality. In this paper, we show that, for any <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="r greater-than-or-equal-to 1"> <mml:semantics> <mml:mrow> <mml:mi>r</mml:mi> <mml:mo>≥<!-- ≥ --></mml:mo> <mml:mn>1</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">r\geq 1</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, there exists a non-elementary Kleinian group whose Jørgensen number is equal to <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="r"> <mml:semantics> <mml:mi>r</mml:mi> <mml:annotation encoding="application/x-tex">r</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. This answers a question posed by Oichi and Sato. We also present our computer generated picture which estimates Jørgensen numbers from above in the diagonal slice of Schottky space.</p>
Umfang: 17-31
ISSN: 1088-4173
DOI: 10.1090/ecgd/331