The diagonal slice of Schottky space

Abstract

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 free and discrete, the resulting manifold is in general a genus-2 handlebody. We study the diagonal slice of the representation variety in which Tr X = Tr Y = Tr XY . Using the symmetry, we are able to compute the Keen-Series pleating rays and thus fully determine the locus of free and discrete groups. 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.

Random Kleinian Groups, II Two Parabolic Generators

“…We can use φ(p/q) for any p/q ∈ Q to get a description of (the interior of) the complement of Bowditch's space. The algorithmic study of this conjecture and related questions is also addressed in [5]. Using these ideas we can estimate the spherical area of the exterior of the Riley slice to be 0.779 × 4π.…”

The realization problem for Jørgensen numbers

“…Remark 2.4. Generating pairs similar to A a and B were studied in [21], and called the singular solid torus.…”

“…See section 5.0.2 (in particular, Remark 5.4 for the sign of the square roots) in [21]. From now on, we consider that x is a complex parameter.…”

“…From now on, we consider that x is a complex parameter. The locus D in the x-plane of discrete and faithful representations was fully determined by computing Keen-Series pleating rays [21]. (Here, "faithful" means that A a , B is isomorphic to A 1 , B as an abstract group.)…”

“…Remark 2.1. Groups similar to P a , Q, R were studied by C. Series, S. P. Tan and the first author in [21].…”