Teichmüller Theory of Bordered Surfaces
We propose the graph description of Teichmüller theory of surfaces with marked points on boundary components (bordered surfaces). Introducing new parameters, we formulate this theory in terms of hyperbolic geometry. We can then describe both classical and quantum theories having the proper number of...
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| Опубліковано в: : | Symmetry, Integrability and Geometry: Methods and Applications |
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| Дата: | 2007 |
| Автор: | |
| Формат: | Стаття |
| Мова: | Англійська |
| Опубліковано: |
Інститут математики НАН України
2007
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| Онлайн доступ: | https://nasplib.isofts.kiev.ua/handle/123456789/147366 |
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| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Цитувати: | Teichmüller Theory of Bordered Surfaces / L.O. Chekhov // Symmetry, Integrability and Geometry: Methods and Applications. — 2007. — Т. 3. — Бібліогр.: 23 назв. — англ. |
Репозитарії
Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1862572527668166656 |
|---|---|
| author | Chekhov, L.O. |
| author_facet | Chekhov, L.O. |
| citation_txt | Teichmüller Theory of Bordered Surfaces / L.O. Chekhov // Symmetry, Integrability and Geometry: Methods and Applications. — 2007. — Т. 3. — Бібліогр.: 23 назв. — англ. |
| collection | DSpace DC |
| container_title | Symmetry, Integrability and Geometry: Methods and Applications |
| description | We propose the graph description of Teichmüller theory of surfaces with marked points on boundary components (bordered surfaces). Introducing new parameters, we formulate this theory in terms of hyperbolic geometry. We can then describe both classical and quantum theories having the proper number of Thurston variables (foliation-shear coordinates), mapping-class group invariance (both classical and quantum), Poisson and quantum algebra of geodesic functions, and classical and quantum braid-group relations. These new algebras can be defined on the double of the corresponding graph related (in a novel way) to a double of the Riemann surface (which is a Riemann surface with holes, not a smooth Riemann surface). We enlarge the mapping class group allowing transformations relating different Teichmüller spaces of bordered surfaces of the same genus, same number of boundary components, and same total number of marked points but with arbitrary distributions of marked points among the boundary components. We describe the classical and quantum algebras and braid group relations for particular sets of geodesic functions corresponding to An and Dn algebras and discuss briefly the relation to the Thurston theory.
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| first_indexed | 2025-11-26T05:07:58Z |
| format | Article |
| fulltext | |
| id | nasplib_isofts_kiev_ua-123456789-147366 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2025-11-26T05:07:58Z |
| publishDate | 2007 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Chekhov, L.O. 2019-02-14T14:45:23Z 2019-02-14T14:45:23Z 2007 Teichmüller Theory of Bordered Surfaces / L.O. Chekhov // Symmetry, Integrability and Geometry: Methods and Applications. — 2007. — Т. 3. — Бібліогр.: 23 назв. — англ. 1815-0659 2000 Mathematics Subject Classification: 37D40; 53C22 https://nasplib.isofts.kiev.ua/handle/123456789/147366 We propose the graph description of Teichmüller theory of surfaces with marked points on boundary components (bordered surfaces). Introducing new parameters, we formulate this theory in terms of hyperbolic geometry. We can then describe both classical and quantum theories having the proper number of Thurston variables (foliation-shear coordinates), mapping-class group invariance (both classical and quantum), Poisson and quantum algebra of geodesic functions, and classical and quantum braid-group relations. These new algebras can be defined on the double of the corresponding graph related (in a novel way) to a double of the Riemann surface (which is a Riemann surface with holes, not a smooth Riemann surface). We enlarge the mapping class group allowing transformations relating different Teichmüller spaces of bordered surfaces of the same genus, same number of boundary components, and same total number of marked points but with arbitrary distributions of marked points among the boundary components. We describe the classical and quantum algebras and braid group relations for particular sets of geodesic functions corresponding to An and Dn algebras and discuss briefly the relation to the Thurston theory. This paper is a contribution to the Vadim Kuznetsov Memorial Issue ‘Integrable Systems and Related Topics’. The author is indebted to V.V. Fock and R.C. Penner for the fruitful discussion on the Oberwolfach Conference on Teichm¨uller spaces, which initiated this work. This work has been partially financially supported by the RFBR Grant No. 05-01-00498, by the Grant for Support of the Leading Scientific Schools 2052.2003.1, by the Program Mathematical Methods of Nonlinear Dynamics, by the ANS Grant “G´eom´etrie et Int´egrabilit´e en Physique Math´ematique” (contract number ANR-05-BLAN-0029-01), and by the European Community through the FP6 Marie Curie RTN ENIGMA (Contract number MRTN-CT-2004-5652). en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Teichmüller Theory of Bordered Surfaces Article published earlier |
| spellingShingle | Teichmüller Theory of Bordered Surfaces Chekhov, L.O. |
| title | Teichmüller Theory of Bordered Surfaces |
| title_full | Teichmüller Theory of Bordered Surfaces |
| title_fullStr | Teichmüller Theory of Bordered Surfaces |
| title_full_unstemmed | Teichmüller Theory of Bordered Surfaces |
| title_short | Teichmüller Theory of Bordered Surfaces |
| title_sort | teichmüller theory of bordered surfaces |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/147366 |
| work_keys_str_mv | AT chekhovlo teichmullertheoryofborderedsurfaces |