Detecting properties from descriptions of groups

Abstract

We consider whether given a simple, finite description of a group in the form of an algorithm, it is possible to algorithmically determine if the corresponding group has some specified property or not. When there is such an algorithm, we say the property is recursively recognizable within some class of descriptions. When there is not, we ask how difficult it is to detect the property in an algorithmic sense.We consider descriptions of two sorts: first, recursive presentations in terms of generators and relators, and second, algorithms for computing the group operation. For both classes of descriptions, we show that a large class of natural algebraic properties, Markov properties, are not recursively recognizable, indeed they are Π 0 2 -hard to detect in recursively presented groups and Π 0 1 -hard to detect in computable groups. These theorems suffice to give a sharp complexity measure for the detection problem of a number of typical group properties, for example, being abelian, torsion-free, orderable. Some properties, like being cyclic, nilpotent, or solvable, are much harder to detect, and we give sharp characterizations of the corresponding detection problems from a number of them.We give special attention to orderability properties, as this was a main motivation at the beginning of this project.

Remarks and problems about algorithmic descriptions of groups

“…The use of algorithms to describe countable groups is just one amongst many means to obtain finite definitions of potentially infinite groups. We can quote here several approaches to describing groups that are commonly used: Note that algorithmic descriptions of groups were already studied in [BCR19], for groups that are not finitely generated. The results of this article justify our focus on finitely generated groups, by showing that almost no global problem is decidable for infinitely generated groups described by algorithms.…”

“…The results of this article justify our focus on finitely generated groups, by showing that almost no global problem is decidable for infinitely generated groups described by algorithms. Note that, in [BCR19], some interesting problems are still raised in the setting of infinitely generated groups, by going past the problem of solvability of different decision problems, and asking more precisely for their location in the Kleene-Mostowski hierarchy.…”

Notes about decidability of exponential equations

“…. , g n , e) ∈ PP[n + 1], where e is the unit of G. The main result of this paper shows that decidability of PP [1] does not imply decidability of PP [2].…”

“…Theorem A. There exists a finitely presented group with decidable PP [1] and undecidable PP [2]. Moreover, this group has decidable conjugacy problem.…”

Computable analysis on the space of marked groups

“…The following well known characterization of orderable groups was already used in [BCR19] in order to study the complexity of the property "being orderable": Proposition 5.17 ([BCR19]). A group G is orderable if and only if for any finite set {a 1 , a 2 , ..., a n } of non-identity elements of G, there are signs ǫ 1 , ..., ǫ n ∈ {−1, 1}, such that the sub-semi-group generated by {a ǫ1 1 , a ǫ2 2 , ..., a ǫn n } does not contain the identity of G.…”