Pants complex, TQFT and hyperbolic geometry
| Autor: | Detcherry, Renaud, Kalfagianni, Efstratia |
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| Přispěvatelé: | HEP, INSPIRE, Institut de Mathématiques de Bourgogne [Dijon] (IMB), Centre National de la Recherche Scientifique (CNRS)-Université de Franche-Comté (UFC), Université Bourgogne Franche-Comté [COMUE] (UBFC)-Université Bourgogne Franche-Comté [COMUE] (UBFC)-Université de Bourgogne (UB) |
| Jazyk: | angličtina |
| Rok vydání: | 2021 |
| Předmět: |
geometry
asymptotic expansion graph theory [PHYS.MPHY]Physics [physics]/Mathematical Physics [math-ph] Geometric Topology (math.GT) [PHYS.MPHY] Physics [physics]/Mathematical Physics [math-ph] field theory: topological Mathematics::Geometric Topology group: representation Mathematics - Geometric Topology SU(2) FOS: Mathematics surface space: noncompact |
| Popis: | We present a coarse perspective on relations of the $SU(2)$-Witten-Reshetikhin-Turaev TQFT, the Weil-Petersson geometry of the Teichm\"uller space, and volumes of hyperbolic 3-manifolds. Using data from the asymptotic expansions of the curve operators in the skein theoretic version of the $SU(2)$-TQFT, as developed by Blanchet, Habegger, Masbaum and Vogel, we define the quantum intersection number between pants decompositions of a closed surface. We show that the quantum intersection number admits two sided bounds in terms of the geometric intersection number and we use it to obtain a metric on the pants graph of surfaces. Using work of Brock we show that the pants graph equipped with this metric is quasi-isometric to the Teichm\"uller space with the Weil-Petersson metric and that the translation length of our metric provides two sided linear bounds on the volume of hyperbolic fibered manifolds. We also obtain a characterization of pseudo-Anosov mapping classes in terms of asymptotics of the quantum intersection number under iteration in the mapping class group and relate these asymptotics with stretch factors. Comment: 38 pages, 6 Figures. Minor changes in introduction, reference updated |
| Databáze: | OpenAIRE |
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