Discreteness is undecidable

Question 1.1. Let G be a connected Lie group and let A = (A1, . . . , Ak) be a finite ordered subset of G. Is the discreteness problem for the subgroup ΓA := 〈A1, . . . , Ak〉 < G decidable? This question, in the case of G = PSL(2,C), was raised, most recently, in the paper [8] by J. Gilman and L. Keen, who noted that “it is a challenging problem that has been investigated for more than a century and is still open.” The decidability problem was solved in the case G = PSL(2,R) by R. Riley [20] and, more efficiently, in the case of 2-generated subgroups, by J. Gilman and B. Maskit [9] and Gilman [6], (cf. [7] for a comparison of the two approaches). To make the general decidability question more precise one has to specify the model of computability. There are several computability models over the real numbers; we refer the reader to [1] and [21] for summaries of these and in-depth treatment of the BSS and the bit-computability approaches respectively. In this paper we address decidability of the discreteness problem in the real-RAM or BSS (which stands for Blum–Shub–Smale) computability model as it is the closest in spirit to the papers by Gilman, Maskit and Keen mentioned above as well as Riley’s work [20].