Computable analysis on the space of marked groups

TL;DR

This paper explores the intersection of computable analysis and group theory, focusing on decision problems for finitely generated groups, and classifies group properties within effective Borel hierarchies.

Contribution

It introduces a systematic study of decision problems in finitely generated groups using computable analysis, including conditions for group properties and classifications within Borel hierarchies.

Findings

Characterizes conditions for solvable word problems in finitely presented groups.

Classifies group properties in effective Borel hierarchies.

Shows the space of marked groups is not computably Polish.

Abstract

We begin the systematic study of decision problems for finitely generated groups given by a solution to their word problem. We relate this to the study of computable analysis on the space of marked groups. We point out that several distinct approaches to computable analysis, some of which are sometimes considered obsolete, yield relevant results. In particular, we give necessary and sufficient conditions in terms of Banach-Mazur computability for the existence of a finitely presented group with solvable word problem but whose subgroups with a certain property cannot be recognized. We classify group properties in different effective Borel hierarchies. For most common group properties, the classical and effective Borel classifications coincide. However, we show that the set of LEF groups is a closed set that is computably a $G_{\delta}$, but not computably closed. Finally, we show that the space of marked groups is a Polish space which is not $\textit{computably Polish}$, because it does not admit a dense and computable sequence. This poses several interesting problems in terms of computable topology. The space of marked groups is the first natural example of this kind.