Can anyone suggest to me some references for studying triangle groups? Especially the existence of finite index subgroups, subgroups isomorphic to fundamental groups of compact surfaces etc.



Reference for triangle groups
It seems that you are interested in “Fenchel’s conjecture” stating that (essentially) all two-orbifolds are finitely (orbifold) covered by surfaces. The case of triangle orbifolds is the hardest, and was solved by Ralph Fox (with, it seems, a few errors). See the 1983 paper by Chau titled [*A note concerning Fox’s paper on Fenchel’s conjecture*](https://doi.org/10.2307/2045442).


