Computability of finite quotients of finitely generated groups

Abstract

We systematically study groups whose marked finite quotients form a recursive set. We give several definitions, and prove basic properties of this class of groups, and in particular emphasize the link between the growth of the depth function and solvability of the word problem. We give examples of infinitely presented groups whose finite quotients can be effectively enumerated. Finally, our main result is that a residually finite group can fail to be recursively presented and still have computable finite quotients, and that, on the other hand, it can have solvable word problem but not have computable finite quotients.

Obstruction to a Higman embedding theorem for residually finite groups with solvable word problem

“…Thus A does not meet x + N Z. The proof of the converse follows closely the proof of (2) given in [11] (which differs from the original proof of Dyson), as one only needs to see that the hypothesis that A is effectively closed in Furstenberg's topology is enough to effectively carry out that proof. As we actually only use the first implication in the proof of Theorem 2, no further details are given here.…”

“…The author already used Dyson's groups to investigate the property of having computable finite quotients ( [11]). What we construct here is a strengthening of the result obtained in that first article, we will include here all definitions but omit the proofs that already appear there.…”

“…What we do not know yet is whether the condition of being recursively presented and effectively residually finite, which is necessary to be a subgroup of a finitely presented residually finite group, is also sufficient. For instance, the strictly stronger condition of having computable finite quotients is not known to be unnecessary (see [11]).…”

“…Fact 3 follows from McKinsey's algorithm. It comes from the fact that a finitely presented group has computable finite quotients, see [11]. Fact 4 is straightforward.…”