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A parabolic representation of the free group    F  2    is one in which the images of both generators are parabolic elements of    P  S  L  (  2  ,  C  )   . Roughly the Riley slice is an open subset    R  ⊂  C    which is a model for the parabolic, discrete and faithful characters of    F  2   . The complement of the closure 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    P  S  L  (  2  ,  C  )   . Recent work of Aimi  et al . 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 representationρ  ∗   :   F  2   →  P  S  L   (  2  ,  C  )whatsoever, any nondiscrete parabolic representation    ρ  0    has an arbitrarily small perturbation    ρ  ε    so thatρ  ε    (   F  2   )contains a conjugate ofρ  ∗    (   F  2   )as a proper subgroup. This implies that if    Γ  ∗    is any nonelementary group generated by two parabolic elements (discrete or otherwise) and    γ  0    is any point in the complement of the Riley slice, then in any neighbourhood of    γ  0    there is a point corresponding to a nonelementary group generated by two parabolics with a conjugate of    Γ  ∗    as a proper subgroup. Using these ideas, we then show the density of nondiscrete parabolic representations with an arbitrarily large number of distinct Nielsen classes with two parabolic generators in the complement of the Riley slice.

Nondiscrete parabolic characters of the free group F 2 : supergroup density and Nielsen classes in the complement of the Riley slice