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

Abstract

We prove that, for a finitely generated residually finite group, having solvable word problem is not a sufficient condition to be a subgroup of a finitely presented residually finite group. The obstruction is given by a residually finite group with solvable word problem for which there is no effective method that allows, given some non-identity element, to find a morphism onto a finite group in which this element has a non-trivial image. We also prove that the depth function of this group grows faster than any recursive function.

Computability of finite quotients of finitely generated groups

“…Because of this, Theorem 1 allows one to ask the following questions: can a residually finite group with solvable word problem have a depth function that grows faster than any recursive function? We solve affirmatively this problem in a follow-up article ( [16]). The obtained residually finite group, although it has solvable word problem, does not embed in a finitely presented residually finite group (because, in [7] again, Bou-Rabee shows that if H is a finitely generated subgroup of a group G, the depth function of H is bounded above by that of G, up to constants).…”

“…In [16], we construct, also using Dyson's groups, a residually finite group G with solvable word problem, that not only does not have CFQ, but that also is not effectively residually finite: there can be no algorithm that, given a non-trivial element w, gives a finite quotient (F, f ) in which the image of w is non-trivial. This is done by constructing a closed subset A of Z that is not effectively closed.…”

An uncountable family of finitely generated residually finite groups

“…Around the same time, Dyson constructed a concrete family of examples of finitely generated residually finite groups with unsolvable word problem [Dys74]. Recently, Rauzy [Rau20] used Dyson's examples to prove that Higman's embedding theorem cannot be extended to the category of finitely generated residually finite groups with solvable word problem.…”

Computable analysis on the space of marked groups

“…In [Rau20] the author constructed a residually finite group with solvable word problem, that is not effectively residually finite. It is then interesting to note that, applied to this group, Proposition 5.6 produces a non-trivial result: the group G is proven to be adherent to the set of finite groups thanks to a sequence of its quotients, which is necessarily not computable, however the proposition then shows that G must also be the limit of an effectively converging sequence of finite groups.…”

“…Note that peculiar instances of this problem naturally arise in our study: in [Rau20], the author has constructed a residually finite group G with solvable word problem with the property that any sequence of its finite quotients that converges to it must be non computable. Thus one cannot apply Markov's Lemma to prove that G cannot be distinguished from its finite quotients.…”