TITLE: Algorithm to test for discrete or quasi-Fuchsian subgroups of PSL(2,C)
QUESTION [14 upvotes]: 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 $\mathrm{PSL}(2,\mathbb{R})$.

REPLY [7 votes]: Indeed, the answer to the discreteness problem turns out to depend on the computability model. In the Real-RAM or BSS (Blum-Shub-Smale) model of computations over the reals, the answer is negative: The discreteness problem (for finitely generated nonelementary subgroups of $PSL(2,C)$) is undecidable, see here, as in the case of the Mandelbrot set $M$ (due to Blum, Shub and Smale). On the other hand, in the bit-computability model, the discreteness problem for finitely generated nonelementary subgroups of $PSL(2,C)$, turns out to be decidable. Interestingly, computability of the Mandelbrot set $M$ in the bit-computability model is still an open problem. However, $M$ is bit-computable, conditional on some major conjectures in holomorphic dynamics (say, local connectivity of $M$). This conditional result is due to Peter Hertling.