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Remarks and problems about algorithmic descriptions of groups

Emmanuel Rauzy

TL;DR

The work develops a unified, lattice- and numbering-based framework for global decision problems in finitely generated groups, bridging local descriptions (word problem) and global descriptions (finite presentations) via multiple computability lenses. It proves Rice–Shapiro-type results for recursive presentations and a Rice theorem for co-recursive presentations, and establishes an algorithmic characterization of finitely presented groups through the equivalence $\nu_{FP} \equiv \nu_{RP} \wedge \nu_{MQA}$, with the compactness of FP groups linked to Scott topology on marked groups. The paper further analyzes marked quotient algorithms, both in the absolute setting and relative to classes of groups, and demonstrates how Marked Quotient content interacts with finite presentations and group varieties, including the lamplighter example. It also discusses Adian–Rabin results as incomplete answers within this framework, outlining how the topology and hierarchy perspectives illuminate undecidability and classification limits in group theory. Overall, the work offers a rigorous, extensible program for algorithmic descriptions of groups beyond traditional finite presentations, with implications for meta-decision problems and the structure of algorithmic group theory.

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.

Remarks and problems about algorithmic descriptions of groups

TL;DR

The work develops a unified, lattice- and numbering-based framework for global decision problems in finitely generated groups, bridging local descriptions (word problem) and global descriptions (finite presentations) via multiple computability lenses. It proves Rice–Shapiro-type results for recursive presentations and a Rice theorem for co-recursive presentations, and establishes an algorithmic characterization of finitely presented groups through the equivalence , with the compactness of FP groups linked to Scott topology on marked groups. The paper further analyzes marked quotient algorithms, both in the absolute setting and relative to classes of groups, and demonstrates how Marked Quotient content interacts with finite presentations and group varieties, including the lamplighter example. It also discusses Adian–Rabin results as incomplete answers within this framework, outlining how the topology and hierarchy perspectives illuminate undecidability and classification limits in group theory. Overall, the work offers a rigorous, extensible program for algorithmic descriptions of groups beyond traditional finite presentations, with implications for meta-decision problems and the structure of algorithmic group theory.

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.

Paper Structure

This paper contains 29 sections, 37 theorems, 22 equations.

Table of Contents

  1. Introduction
  2. Undecidability results as non-classification results
  3. General setting: numberings of marked groups
  4. Decision problems with respect to local descriptions
  5. Finite presentations and marked quotient algorithms
  6. Lattice of $k$-marked groups and scott topology
  7. The lattice of $k$-marked groups
  8. Scott topology of the lattice of $k$-marked groups
  9. Lattice of numbering types
  10. Definitions of the lattice of equivalence classes of numberings
  11. Semi-decidable sets, Ershov topology
  12. Numberings associated to finite presentations, recursive presentations, etc
  13. Examples where the explicit use of numberings is beneficial
  14. First example: recursive presentations
  15. Second example: decidability modulo the word problem and Banch-Mazur computability
  16. ...and 14 more sections

Key Result

Theorem 1

There exists an algorithm that, given as input a presentation for a group $G$ and a solution to the word problem in $G$, determines whether or not $G$ is free.

Theorems & Definitions (81)

  • Theorem 1: Groves2009
  • Lemma 3
  • Theorem 4: Rice-Shapiro theorem for recursive presentations
  • Corollary 5: Rice theorem for recursive presentations
  • Definition 6: Marked quotient problem
  • Theorem 7: Algorithmic characterization of finitely presented groups
  • Definition 8: Relative marked quotient problem
  • Proposition 10
  • proof
  • Proposition 11
  • ...and 71 more