Moduli spaces of local systems and higher Teichmüller theory
Identifieur interne : 000D72 ( Istex/Corpus ); précédent : 000D71; suivant : 000D73Moduli spaces of local systems and higher Teichmüller theory
Auteurs : Vladimir Fock ; Alexander GoncharovSource :
- Publications Mathématiques de l'Institut des Hautes Études Scientifiques [ 0073-8301 ] ; 2006-06-01.
Abstract
Abstract: Let G be a split semisimple algebraic group over Q with trivial center. Let S be a compact oriented surface, with or without boundary. We define positive representations of the fundamental group of S to G(R), construct explicitly all positive representations, and prove that they are faithful, discrete, and positive hyperbolic; the moduli space of positive representations is a topologically trivial open domain in the space of all representations. When S have holes, we defined two moduli spaces closely related to the moduli spaces of G-local systems on S. We show that they carry a lot of interesting structures. In particular we define a distinguished collection of coordinate systems, equivariant under the action of the mapping class group of S. We prove that their transition functions are subtraction free. Thus we have positive structures on these moduli spaces. Therefore we can take their points with values in any positive semifield. Their positive real points provide the two higher Teichmüller spaces related to G and S, while the points with values in the tropical semifields provide the lamination spaces. We define the motivic avatar of the Weil–Petersson form for one of these spaces. It is related to the motivic dilogarithm.
Url:
DOI: 10.1007/s10240-006-0039-4
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Let S be a compact oriented surface, with or without boundary. We define positive representations of the fundamental group of S to G(R), construct explicitly all positive representations, and prove that they are faithful, discrete, and positive hyperbolic; the moduli space of positive representations is a topologically trivial open domain in the space of all representations. When S have holes, we defined two moduli spaces closely related to the moduli spaces of G-local systems on S. We show that they carry a lot of interesting structures. In particular we define a distinguished collection of coordinate systems, equivariant under the action of the mapping class group of S. We prove that their transition functions are subtraction free. Thus we have positive structures on these moduli spaces. Therefore we can take their points with values in any positive semifield. Their positive real points provide the two higher Teichmüller spaces related to G and S, while the points with values in the tropical semifields provide the lamination spaces. We define the motivic avatar of the Weil–Petersson form for one of these spaces. It is related to the motivic dilogarithm.</div></front></TEI><istex><corpusName>springer-journals</corpusName><author><json:item><name>Vladimir Fock</name><affiliations><json:string>ITEP, B. 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We define positive representations of the fundamental group of S to G(R), construct explicitly all positive representations, and prove that they are faithful, discrete, and positive hyperbolic; the moduli space of positive representations is a topologically trivial open domain in the space of all representations. When S have holes, we defined two moduli spaces closely related to the moduli spaces of G-local systems on S. We show that they carry a lot of interesting structures. In particular we define a distinguished collection of coordinate systems, equivariant under the action of the mapping class group of S. We prove that their transition functions are subtraction free. Thus we have positive structures on these moduli spaces. Therefore we can take their points with values in any positive semifield. Their positive real points provide the two higher Teichmüller spaces related to G and S, while the points with values in the tropical semifields provide the lamination spaces. We define the motivic avatar of the Weil–Petersson form for one of these spaces. 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Their positive real points provide the two higher Teichmüller spaces related to G and S, while the points with values in the tropical semifields provide the lamination spaces. We define the motivic avatar of the Weil–Petersson form for one of these spaces. 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Cheremushkinskaya 25</Street><Postcode>117259</Postcode><City>Moscow</City><Country Code="RU">Russia</Country></OrgAddress></Affiliation><Affiliation ID="Aff2"><OrgDivision>Department of Mathematics</OrgDivision><OrgName>Brown University</OrgName><OrgAddress><City>Providence</City><State>RI</State><Postcode>02912</Postcode><Country Code="US">USA</Country></OrgAddress></Affiliation></AuthorGroup><Abstract Language="En" ID="Abs1"><Heading>Abstract</Heading><Para>Let G be a split semisimple algebraic group over <Emphasis Type="Bold">Q</Emphasis> with trivial center. Let S be a compact oriented surface, with or without boundary. We define <Emphasis Type="Italic">positive</Emphasis> representations of the fundamental group of S to G(<Emphasis Type="Bold">R</Emphasis>), construct explicitly all positive representations, and prove that they are faithful, discrete, and positive hyperbolic; the moduli space of positive representations is a topologically trivial open domain in the space of all representations. When S have holes, we defined two moduli spaces closely related to the moduli spaces of G-local systems on S. We show that they carry a lot of interesting structures. In particular we define a distinguished collection of coordinate systems, equivariant under the action of the mapping class group of S. We prove that their transition functions are subtraction free. Thus we have positive structures on these moduli spaces. Therefore we can take their points with values in any positive semifield. Their positive real points provide the two higher Teichmüller spaces related to G and S, while the points with values in the tropical semifields provide the lamination spaces. We define the motivic avatar of the Weil–Petersson form for one of these spaces. It is related to the motivic dilogarithm. </Para></Abstract></ArticleHeader><NoBody></NoBody></Article></Issue></Volume></Journal></Publisher></istex:document></istex:metadataXml><mods version="3.6"><titleInfo lang="en"><title>Moduli spaces of local systems and higher Teichmüller theory</title></titleInfo><titleInfo type="alternative" contentType="CDATA" lang="en"><title>Moduli spaces of local systems and higher Teichmüller theory</title></titleInfo><name type="personal" displayLabel="corresp"><namePart type="given">Vladimir</namePart><namePart type="family">Fock</namePart><affiliation>ITEP, B. Cheremushkinskaya 25, 117259, Moscow, Russia</affiliation><affiliation>E-mail: fock@math.brown.edu</affiliation><role><roleTerm type="text">author</roleTerm></role></name><name type="personal" displayLabel="corresp"><namePart type="given">Alexander</namePart><namePart type="family">Goncharov</namePart><affiliation>Department of Mathematics, Brown University, 02912, Providence, RI, USA</affiliation><affiliation>E-mail: sasha@math.brown.edu</affiliation><role><roleTerm type="text">author</roleTerm></role></name><typeOfResource>text</typeOfResource><genre type="research-article" displayLabel="OriginalPaper" authority="ISTEX" authorityURI="https://content-type.data.istex.fr" valueURI="https://content-type.data.istex.fr/ark:/67375/XTP-1JC4F85T-7">research-article</genre><originInfo><publisher>Springer-Verlag; www.springer.de</publisher><place><placeTerm type="text">Berlin/Heidelberg</placeTerm></place><dateCreated encoding="w3cdtf">2004-06-10</dateCreated><dateIssued encoding="w3cdtf">2006-06-01</dateIssued><dateIssued encoding="w3cdtf">2006</dateIssued><copyrightDate encoding="w3cdtf">2006</copyrightDate></originInfo><language><languageTerm type="code" authority="rfc3066">en</languageTerm><languageTerm type="code" authority="iso639-2b">eng</languageTerm></language><abstract lang="en">Abstract: Let G be a split semisimple algebraic group over Q with trivial center. Let S be a compact oriented surface, with or without boundary. We define positive representations of the fundamental group of S to G(R), construct explicitly all positive representations, and prove that they are faithful, discrete, and positive hyperbolic; the moduli space of positive representations is a topologically trivial open domain in the space of all representations. When S have holes, we defined two moduli spaces closely related to the moduli spaces of G-local systems on S. We show that they carry a lot of interesting structures. In particular we define a distinguished collection of coordinate systems, equivariant under the action of the mapping class group of S. We prove that their transition functions are subtraction free. Thus we have positive structures on these moduli spaces. Therefore we can take their points with values in any positive semifield. Their positive real points provide the two higher Teichmüller spaces related to G and S, while the points with values in the tropical semifields provide the lamination spaces. We define the motivic avatar of the Weil–Petersson form for one of these spaces. 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