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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
Forum Mathematicum – de Gruyter
Published: Oct 1, 2003
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