Let $\Gamma = \pi_1(S)$ denote the fundamental group of a compact surface $S$ of genus $g>1$.

Given a representation $\rho : \Gamma \to \mathrm{PSL}(2,\mathbb{C})$, specified by matrix representatives for the images of a fixed generating set, is there an algorithm to answer either of the following questions?

1) Is the image $\rho(\Gamma)$ a quasi-Fuchsian group?

2) Is the image $\rho(\Gamma)$ discrete?

There are a number of related situations where I am aware of algorithms of this type, but all are limited to special classes of two-generator subgroups of $\mathrm{PSL}(2,\mathbb{C})$. For example:

For two-generator subgroups of $\mathrm{PSL}(2,\mathbb{R})$ there is the Gilman-Maskit algorithm which tests for discreteness. There are some related sufficient conditions for discreteness in the two-generator case in $\mathrm{PSL}(2,\mathbb{C})$.

For punctured torus groups (i.e. representations of $\mathbb{F}_2$ where $abab^{-1}$ maps to a parabolic element) in $\mathrm{PSL}(2,\mathbb{C})$ there is a method of Komori, Sugawa, Wada, and Yamashita based on simultaneously testing Jorgensen's inequality (attempting to find a certificate that the group is not discrete) while also trying to find a Ford fundamental domain (of a type that would give a certificate that the group is quasi-Fuchsian).

Also in the punctured torus case, Bowditch has a conjectural characterization of quasi-Fuchsian groups in terms of a certain subset of the infinite trivalent tree of "generating triples", which is easy to test algorithmically. As in the previous case this can be combined with Jorgensen's inequality to get a heuristic test for discreteness.

Based on these cases I would especially like to know about methods for discreteness or quasi-Fuchsian testing that apply in the compact surface case without assuming that the representation maps into $\mathrm{PSL}(2,\mathbb{R})$.

aminterested in the closed surface case. $\endgroup$