The diagonal slice of Schottky space

Series, Caroline¹ ; Tan, Ser² ; Yamashita, Yasushi³

¹Mathematics Institute, University of Warwick, Leeman Building, Coventry, CV4 7AL, United Kingdom
²Department of Mathematics, National University of Singapore, Block S17, 10 Lower Kent Ridge Rd., Singapore 119076
³Department of Information and Computer Sciences, Nara Women’s University, Kitauoyanishi-machi, Nara 630-8506, Japan

Algebraic and Geometric Topology, Tome 17 (2017) no. 4, pp. 2239-2282

Résumé

An irreducible representation of the free group on two generators X,Y into SL(2, ℂ) is determined up to conjugation by the traces of X,Y and XY . If the representation is faithful and discrete, the resulting manifold is in general a genus-2 handlebody. We study the diagonal slice of the representation variety in which TrX = TrY = TrXY . Using the symmetry, we are able to compute the Keen–Series pleating rays and thus fully determine the locus of faithful discrete representations. We also computationally determine the “Bowditch set” consisting of those parameter values for which no primitive elements in 〈X,Y 〉 have traces in [−2,2], and at most finitely many primitive elements have traces with absolute value at most 2. The graphics make clear that this set is both strictly larger than, and significantly different from, the discreteness locus.

Détail

DOI : 10.2140/agt.2017.17.2239

Classification : 30F40, 57M50
Keywords: Schottky group, nondiscrete group, primitive element, Bowditch condition

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