Discreteness is undecidable

Michael Kapovich
2016 International journal of algebra and computation  
We prove that the discreteness problem for 2-generated nonelementary subgroups of SLp2, Cq is undecidable in the BSS computability model. This paper is motivated by the following basic question: Question 1. Let G be a connected Lie group and let A " pA 1 , . . . , A k q be a finite ordered subset of G. Is the discreteness problem for the subgroup Γ A :" xA 1 , . . . , A k y ă G decidable? This question, in the case of G " P SLp2, Cq, was raised, most recently, in the paper [8] by J. Gilman and
more » ... ] 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 " P SLp2, Rq 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 bitcomputability 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] . We will address decidability of the discreteness problem in the bitcomputabulity model in another paper, [13] . Remark 2. We refer the reader to the paper by J. Gilman in [7] , where several (semi)algorithms for the discreteness problem in P SLp2, Rq and P SLp2, Cq in different computability models, including the BSS model, are compared. Briefly, computations in the BSS model over the real numbers are performed by a BSS machine, which is an analogue of a Turing machine except that a BSS machine can store finite lists of real numbers and do elementary algebraic and order operation with real numbers: Such a machine can add, subtract, multiply and divide, as well as verify inequalities and equalities a ă b, a " b for real numbers. (BSS machines are also defined for computations in other rings, but, in this paper we will use only real numbers.) We refer to [1] for the details. A subset E Ă R n is BSS-semicomputable (or the membership problem for E is BSS-semidecidable) if E is the halting set of a BSS machine: There exists a BSS machine which, given
doi:10.1142/s0218196716500193 fatcat:vaishpgmc5emregspi2glpavau