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 \textit{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, \emph{Markov properties}, are not recursively recognizable, indeed they are $\Pi^0_2$-hard to detect in recursively presented groups and $\Pi^0_1$-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.