The diagonal slice of Schottky space

TL;DR

This paper investigates the diagonal slice of Schottky space by analyzing specific representations of the free group on two generators into SL(2,C), identifying conditions for discreteness and freeness, and computing the Bowditch set with visualizations.

Contribution

It introduces a detailed study of the diagonal slice of Schottky space, locating free and discrete groups, and computes the Bowditch set with novel graphical insights.

Findings

Identifies conditions for free, discrete representations in the diagonal slice.

Computes the Bowditch set and highlights differences from quasifuchsian groups.

Uses symmetry and pleating rays to analyze the structure of these representations.

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. We study the diagonal slice of representations for which X,Y and XY have equal trace. Using the three-fold symmetry and Keen-Series pleating rays we locate those groups which are free and discrete, in which case the resulting hyperbolic manifold is a genus-2 handlebody. We also compute the Bowditch set, consisting of those representations for which no primitive elements in the group generated by X,Y are parabolic or elliptic, and at most finitely many have trace with absolute value at most 2. In contrast to the quasifuchsian punctured torus groups originally studied by Bowditch, computer graphics show that this set is significantly different from the discreteness locus.