Computable analysis on the space of marked groups

Abstract

We investigate decision problems for groups described by word problem algorithms. We show that this corresponds to the study of computable analysis on the space of marked groups, and point out several results of computable analysis that can be directly applied to obtain group theoretical results. However, we prove that the space of marked groups is a Polish space that is not effectively Polish, because of this, many of the most important results of computable analysis cannot be applied to the space of marked groups. This includes the Kreisel-Lacombe-Schoenfield-Ceitin Theorem and different theorems of Moschovakis. The space of marked groups will thus be an interesting object of study from the point of view of computable analysis alone, providing a natural space on which to test results that could apply to spaces that are not effectively Polish.