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Exceptional discrete mapping class group orbits in moduli spaces

Exceptional discrete mapping class group orbits in moduli spaces Abstract. Let M be a four-holed sphere and G the mapping class group of M fixing qM. The group G acts on the space MB ðSUð2ÞÞ of SUð2Þ-gauge equivalence classes of flat SUð2Þconnections on M with fixed holonomy on qM. We give examples of flat SUð2Þ-connections whose holonomy groups are dense in SUð2Þ, but whose G-orbits are discrete in MB ðSUð2ÞÞ. This phenomenon does not occur for surfaces with genus greater than zero. 1991 Mathematics Subject Classification: 57M05, 54H20. 1 Introduction Let M be a Riemann surface of genus g with n boundary components (circles). Let fg1 ; g2 ; . . . ; gn g H p1 ðM Þ be the elements in the fundamental group corresponding to these n boundary components. Assign to each gi a conjugacy class Bi H SUð2Þ and let B ¼ fB1 ; B2 ; . . . ; Bn g; H ¼ fr A Homðp1 ðM Þ; SUð2ÞÞ : rðgi Þ A Bi ; 1 a i a ng: B A conjugacy class in SUð2Þ is determined by its trace which is in ½À2; 2. Hence we might consider B as an element in ½À2; 2 n . The group SUð2Þ http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Forum Mathematicum de Gruyter

Exceptional discrete mapping class group orbits in moduli spaces

Forum Mathematicum , Volume 15 (6) – Oct 1, 2003

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Publisher
de Gruyter
Copyright
Copyright © 2003 by Walter de Gruyter GmbH & Co. KG
ISSN
0933-7741
eISSN
1435-5337
DOI
10.1515/form.2003.048
Publisher site
See Article on Publisher Site

Abstract

Abstract. Let M be a four-holed sphere and G the mapping class group of M fixing qM. The group G acts on the space MB ðSUð2ÞÞ of SUð2Þ-gauge equivalence classes of flat SUð2Þconnections on M with fixed holonomy on qM. We give examples of flat SUð2Þ-connections whose holonomy groups are dense in SUð2Þ, but whose G-orbits are discrete in MB ðSUð2ÞÞ. This phenomenon does not occur for surfaces with genus greater than zero. 1991 Mathematics Subject Classification: 57M05, 54H20. 1 Introduction Let M be a Riemann surface of genus g with n boundary components (circles). Let fg1 ; g2 ; . . . ; gn g H p1 ðM Þ be the elements in the fundamental group corresponding to these n boundary components. Assign to each gi a conjugacy class Bi H SUð2Þ and let B ¼ fB1 ; B2 ; . . . ; Bn g; H ¼ fr A Homðp1 ðM Þ; SUð2ÞÞ : rðgi Þ A Bi ; 1 a i a ng: B A conjugacy class in SUð2Þ is determined by its trace which is in ½À2; 2. Hence we might consider B as an element in ½À2; 2 n . The group SUð2Þ

Journal

Forum Mathematicumde Gruyter

Published: Oct 1, 2003

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