Remarks and problems about algorithmic descriptions of groups

TL;DR

This paper develops a comprehensive framework for studying decision problems in finitely generated groups using the lattice of numberings of isomorphism classes, establishing key theorems and characterizations.

Contribution

It introduces a lattice-based framework, proves Rice and Rice-Shapiro theorems for various presentations, and characterizes finitely presentable groups via semi-decidability of specific decision problems.

Findings

Established Rice and Rice-Shapiro Theorems for recursive presentations

Provided an algorithmic characterization of finitely presentable groups

Discussed limitations of the Adian-Rabin Theorem

Abstract

Motivated by a theorem of Groves and Wilton, we propose the study of the lattice of numberings of isomorphism classes of marked groups as a rigorous and comprehensive framework to study global decision problems for finitely generated groups. We establish the Rice and Rice-Shapiro Theorems for recursive presentations, and establish similar results for co-recursive presentations. We give an algorithmic characterization of finitely presentable groups in terms of semi-decidability of two decision problems: the word problem and the marked quotient problem, which we introduce. We explain how this result can be used to define algorithmic generalizations of finite presentations. Finally, we discuss how the Adian-Rabin Theorem provides incomplete answers in several respects.