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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: | , |
| 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: |
| author_facet |
Yamashita, Yasushi Yamazaki, Ryosuke Yamashita, Yasushi Yamazaki, Ryosuke |
|---|---|
| author |
Yamashita, Yasushi Yamazaki, Ryosuke |
| spellingShingle |
Yamashita, Yasushi Yamazaki, Ryosuke Conformal Geometry and Dynamics of the American Mathematical Society The realization problem for Jørgensen numbers Geometry and Topology |
| author_sort |
yamashita, yasushi |
| spelling |
Yamashita, Yasushi Yamazaki, Ryosuke 1088-4173 American Mathematical Society (AMS) Geometry and Topology http://dx.doi.org/10.1090/ecgd/331 <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> The realization problem for Jørgensen numbers Conformal Geometry and Dynamics of the American Mathematical Society |
| doi_str_mv |
10.1090/ecgd/331 |
| facet_avail |
Online Free |
| finc_class_facet |
Mathematik |
| format |
ElectronicArticle |
| fullrecord |
blob:ai-49-aHR0cDovL2R4LmRvaS5vcmcvMTAuMTA5MC9lY2dkLzMzMQ |
| id |
ai-49-aHR0cDovL2R4LmRvaS5vcmcvMTAuMTA5MC9lY2dkLzMzMQ |
| institution |
DE-L229 DE-D275 DE-Bn3 DE-Brt1 DE-Zwi2 DE-D161 DE-Gla1 DE-Zi4 DE-15 DE-Pl11 DE-Rs1 DE-105 DE-14 DE-Ch1 |
| imprint |
American Mathematical Society (AMS), 2019 |
| imprint_str_mv |
American Mathematical Society (AMS), 2019 |
| issn |
1088-4173 |
| issn_str_mv |
1088-4173 |
| language |
English |
| mega_collection |
American Mathematical Society (AMS) (CrossRef) |
| match_str |
yamashita2019therealizationproblemforjorgensennumbers |
| publishDateSort |
2019 |
| publisher |
American Mathematical Society (AMS) |
| recordtype |
ai |
| record_format |
ai |
| series |
Conformal Geometry and Dynamics of the American Mathematical Society |
| source_id |
49 |
| title |
The realization problem for Jørgensen numbers |
| title_unstemmed |
The realization problem for Jørgensen numbers |
| title_full |
The realization problem for Jørgensen numbers |
| title_fullStr |
The realization problem for Jørgensen numbers |
| title_full_unstemmed |
The realization problem for Jørgensen numbers |
| title_short |
The realization problem for Jørgensen numbers |
| title_sort |
the realization problem for jørgensen numbers |
| topic |
Geometry and Topology |
| url |
http://dx.doi.org/10.1090/ecgd/331 |
| publishDate |
2019 |
| physical |
17-31 |
| description |
<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> |
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Conformal Geometry and Dynamics of the American Mathematical Society |
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1792326014796300288 |
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2024-03-01T12:14:44.37Z |
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url_ver=Z39.88-2004&ctx_ver=Z39.88-2004&ctx_enc=info%3Aofi%2Fenc%3AUTF-8&rfr_id=info%3Asid%2Fvufind.svn.sourceforge.net%3Agenerator&rft.title=The+realization+problem+for+J%C3%B8rgensen+numbers&rft.date=2019-02-25&genre=article&issn=1088-4173&volume=23&issue=2&spage=17&epage=31&pages=17-31&jtitle=Conformal+Geometry+and+Dynamics+of+the+American+Mathematical+Society&atitle=The+realization+problem+for+J%C3%B8rgensen+numbers&aulast=Yamazaki&aufirst=Ryosuke&rft_id=info%3Adoi%2F10.1090%2Fecgd%2F331&rft.language%5B0%5D=eng |
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| _version_ | 1792326014796300288 |
| author | Yamashita, Yasushi, Yamazaki, Ryosuke |
| author_facet | Yamashita, Yasushi, Yamazaki, Ryosuke, Yamashita, Yasushi, Yamazaki, Ryosuke |
| author_sort | yamashita, yasushi |
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| container_title | Conformal Geometry and Dynamics of the American Mathematical Society |
| container_volume | 23 |
| description | <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> |
| doi_str_mv | 10.1090/ecgd/331 |
| facet_avail | Online, Free |
| finc_class_facet | Mathematik |
| format | ElectronicArticle |
| format_de105 | Article, E-Article |
| format_de14 | Article, E-Article |
| format_de15 | Article, E-Article |
| format_de520 | Article, E-Article |
| format_de540 | Article, E-Article |
| format_dech1 | Article, E-Article |
| format_ded117 | Article, E-Article |
| format_degla1 | E-Article |
| format_del152 | Buch |
| format_del189 | Article, E-Article |
| format_dezi4 | Article |
| format_dezwi2 | Article, E-Article |
| format_finc | Article, E-Article |
| format_nrw | Article, E-Article |
| geogr_code | not assigned |
| geogr_code_person | not assigned |
| id | ai-49-aHR0cDovL2R4LmRvaS5vcmcvMTAuMTA5MC9lY2dkLzMzMQ |
| imprint | American Mathematical Society (AMS), 2019 |
| imprint_str_mv | American Mathematical Society (AMS), 2019 |
| institution | DE-L229, DE-D275, DE-Bn3, DE-Brt1, DE-Zwi2, DE-D161, DE-Gla1, DE-Zi4, DE-15, DE-Pl11, DE-Rs1, DE-105, DE-14, DE-Ch1 |
| issn | 1088-4173 |
| issn_str_mv | 1088-4173 |
| language | English |
| last_indexed | 2024-03-01T12:14:44.37Z |
| match_str | yamashita2019therealizationproblemforjorgensennumbers |
| mega_collection | American Mathematical Society (AMS) (CrossRef) |
| physical | 17-31 |
| publishDate | 2019 |
| publishDateSort | 2019 |
| publisher | American Mathematical Society (AMS) |
| record_format | ai |
| recordtype | ai |
| series | Conformal Geometry and Dynamics of the American Mathematical Society |
| source_id | 49 |
| spelling | Yamashita, Yasushi Yamazaki, Ryosuke 1088-4173 American Mathematical Society (AMS) Geometry and Topology http://dx.doi.org/10.1090/ecgd/331 <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> The realization problem for Jørgensen numbers Conformal Geometry and Dynamics of the American Mathematical Society |
| spellingShingle | Yamashita, Yasushi, Yamazaki, Ryosuke, Conformal Geometry and Dynamics of the American Mathematical Society, The realization problem for Jørgensen numbers, Geometry and Topology |
| title | The realization problem for Jørgensen numbers |
| title_full | The realization problem for Jørgensen numbers |
| title_fullStr | The realization problem for Jørgensen numbers |
| title_full_unstemmed | The realization problem for Jørgensen numbers |
| title_short | The realization problem for Jørgensen numbers |
| title_sort | the realization problem for jørgensen numbers |
| title_unstemmed | The realization problem for Jørgensen numbers |
| topic | Geometry and Topology |
| url | http://dx.doi.org/10.1090/ecgd/331 |