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Computable analysis on the space of marked groups

Emmanuel Rauzy

TL;DR

This work links decision problems for finitely generated groups with a solved word problem to computable analysis on the space of marked groups. It develops and compares three computability frameworks (Type 2, Markov, Banach-Mazur) and shows how they yield complementary decidability/undecidability results, often via Kolmogorov complexity arguments. The paper proves that the space of marked groups is a Polish space that is not computably Polish, and it classifies many group properties within effective Borel hierarchies, revealing cases where classical and effective classifications coincide as well as notable exceptions (e.g., LEF groups). It also provides significant results about subgroups of finitely presented groups with solvable word problem, using Miller-type constructions and Higman-Clapham-Valiev embeddings, illustrating deep interactions between computability, topology, and group theory with implications for computable topology and descriptive set theory in algebraic contexts.

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 $G_δ$, 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.

Computable analysis on the space of marked groups

TL;DR

This work links decision problems for finitely generated groups with a solved word problem to computable analysis on the space of marked groups. It develops and compares three computability frameworks (Type 2, Markov, Banach-Mazur) and shows how they yield complementary decidability/undecidability results, often via Kolmogorov complexity arguments. The paper proves that the space of marked groups is a Polish space that is not computably Polish, and it classifies many group properties within effective Borel hierarchies, revealing cases where classical and effective classifications coincide as well as notable exceptions (e.g., LEF groups). It also provides significant results about subgroups of finitely presented groups with solvable word problem, using Miller-type constructions and Higman-Clapham-Valiev embeddings, illustrating deep interactions between computability, topology, and group theory with implications for computable topology and descriptive set theory in algebraic contexts.

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 , but not computably closed. Finally, we show that the space of marked groups is a Polish space which is not , 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.

Paper Structure

This paper contains 77 sections, 116 theorems, 65 equations, 1 table.

Table of Contents

  1. Introduction
  2. Groups given by a solution to their word problem
  3. Choosing a framework
  4. Relationship between the three frameworks
  5. Partial conclusion
  6. Representation and numbering associated to word problems
  7. Topology of the space of marked groups
  8. A non-computably Polish space
  9. Continuity problems on the space of marked groups
  10. Markov's Lemma
  11. Effective Borel hierarchies
  12. Differences between the Borel hierarchy and its effective counterpart
  13. Contents of this paper
  14. The topological space of marked groups
  15. Definitions
  16. ...and 62 more sections

Key Result

Theorem 1.1

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 (217)

  • Theorem 1.1: Groves2009
  • Claim 1.2
  • Theorem 1.3: Kreisel-Lacombe-Schoenfield, KLS57
  • Theorem A
  • Lemma 1.4
  • Lemma 1.5
  • Theorem 1.6: Boone-Novikov, reformulated
  • Theorem 1.7: Boone-Rogers, reformulated
  • Theorem B
  • Theorem C
  • ...and 207 more