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The diagonal slice of Schottky space

Caroline Series, Ser Peow Tan and Yasushi Yamashita

Algebraic & Geometric Topology 17 (2017) 2239–2282

An irreducible representation of the free group on two generators X,Y into SL(2, ) 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.

Schottky group, nondiscrete group, primitive element, Bowditch condition
Mathematical Subject Classification 2010
Primary: 30F40
Secondary: 57M50
Received: 17 April 2016
Revised: 17 October 2016
Accepted: 31 October 2016
Published: 3 August 2017
Caroline Series
Mathematics Institute
University of Warwick
Leeman Building
United Kingdom
Ser Peow Tan
Department of Mathematics
National University of Singapore
Block S17, 10 Lower Kent Ridge Rd.
Singapore 119076
Yasushi Yamashita
Department of Information and Computer Sciences
Nara Women’s University
Kitauoyanishi-machi, Nara 630-8506