A parabolic representation of the free group $F_2$ is one in which the images of both generators are parabolic elements of $PSL(2,\mathbb{C})$. Roughly the Riley slice is an open subset $\mathcal{R}\subset \mathbb{C}$ 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 $PSL(2,\mathbb{C})$. 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 representation ${\rho }_{\ast }:F_2\to PSL(2,\mathbb{C})$ whatsoever, any nondiscrete parabolic representation ${\rho }_0$ has an arbitrarily small perturbation ${\rho }_{\epsilon }$ so that ${\rho }_{\epsilon }(F_2)$ contains a conjugate of ${\rho }_{\ast }(F_2)$ as a proper subgroup. This implies that if ${\mathrm{\Gamma }}_{\ast }$ 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 ${\gamma }_0$ there is a point corresponding to a nonelementary group generated by two parabolics with a conjugate of ${\mathrm{\Gamma }}_{\ast }$ 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.

中文翻译：

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自由群 F2 的非离散抛物线特征：Riley 片补中的超群密度和 Nielsen 类

自由群的抛物线表示 $F_2$ 是一种其中两个生成器的图像都是抛物线元素的 $磷秒升(2,\mathbb{C})$. 大致上，莱利切片是一个开放子集$\mathcal{电阻}\subset \mathbb{C}$ 这是抛物线的、离散的和忠实的特征的模型 $F_2$. Riley 切片的闭包的补是一个有界 Jordan 域，其中有孤立的点，仅在边界处累积，对应于刚性子群的抛物线离散和忠实表示$磷秒升(2,\mathbb{C})$. Aimi等人的近期工作。已经拓扑地确定了所有这些组。在这里，我们给出了非离散表示的第一个确定的实质性性质，并证明了一个超群密度定理：给定任何不可约的抛物线表示${\rho }_{＊}：F_2\to 磷秒升(2,\mathbb{C})$ 无论如何，任何非离散的抛物线表示 ${\rho }_0$ 有一个任意小的扰动 ${\rho }_{\epsilon }$ 以便 ${\rho }_{\epsilon }(F_2)$ 包含一个共轭 ${\rho }_{＊}(F_2)$作为适当的子群。这意味着如果${\mathrm{\Gamma }}_{＊}$ 是由两个抛物线元素（离散或其他）生成的任何非初等群，并且 ${\gamma }_0$ 是 Riley 切片的补中的任意点，然后在 ${\gamma }_0$ 有一个点对应于由两个共轭的抛物线产生的非初等群 ${\mathrm{\Gamma }}_{＊}$作为适当的子群。使用这些想法，然后我们展示了具有任意大量不同 Nielsen 类的非离散抛物线表示的密度，其中两个抛物线生成器位于 Riley 切片的补充中。