Nondiscrete parabolic characters of the free group F2: supergroup density and Nielsen classes in the complement of the Riley slice

Abstract

A parabolic representation of the free group $F_2$ is one in which the images of both generators are parabolic elements of $PSL(2,\IC)$. The Riley slice is a closed subset ${\cal R}\subset \IC$ which is a model for the parabolic, discrete and faithful characters of $F_2$. The complement of the Riley slice is a bounded Jordan domain within which there are isolated points, accumulating only at the boundary, corresponding to parabolic discrete and faithful representations of rigid subgroups of $PSL(2,\IC)$. Recent work of Aimi, Akiyoshi, Lee, Oshika, Parker, Lee, Sakai, Sakuma \& Yoshida, have topologically identified all these groups. Here we give the first identified substantive properties of the nondiscrete representations and prove a supergroup density theorem: given any irreducible parabolic representation $\rho_*:F_2\to PSL(2,\IC)$ whatsoever, any non-discrete parabolic representation $\rho_0$ has an arbitrarily small perturbation $\rho_\epsilon$ so that $\rho_\epsilon(F_2)$ contains a conjugate of $\rho_*(F_2)$ as a proper subgroup. This implies that if $\Gamma_*$ is any nonelementary group generated by two parabolic elements (discrete or otherwise) and $\gamma_0$ is any point in the complement of the Riley slice, then in any neighbourhood of $\gamma$ there is a point corresponding to a nonelementary group generated by two parabolics with a conjugate of $\Gamma_*$ as a proper subgroup. Using these ideas we then show that there are nondiscrete parabolic representations with an arbitrarily large number of distinct Nielsen classes of parabolic generators.