Algebraic & Geometric Topology

The diagonal slice of Schottky space

Caroline Series, Ser Tan, and Yasushi Yamashita

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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.

Article information

Algebr. Geom. Topol., Volume 17, Number 4 (2017), 2239-2282.

Received: 17 April 2016
Revised: 17 October 2016
Accepted: 31 October 2016
First available in Project Euclid: 16 November 2017

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 30F40: Kleinian groups [See also 20H10]
Secondary: 57M50: Geometric structures on low-dimensional manifolds

Schottky group nondiscrete group primitive element Bowditch condition


Series, Caroline; Tan, Ser; Yamashita, Yasushi. The diagonal slice of Schottky space. Algebr. Geom. Topol. 17 (2017), no. 4, 2239--2282. doi:10.2140/agt.2017.17.2239.

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