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On the word problem for just infinite groups

Alexey Talambutsa

Abstract

We establish several results on the word problem for just infinite groups. First, for finitely generated just infinite groups we show that the word problem is uniformly decidable for presentations with recursively enumerable sets of relations. Our proof does not use the Wilson--Grigorchuk theorem on the classification of just infinite groups and proceeds directly from the definition, using ideas from classical results on decidability of the word problem: Kuznetsov's theorem and Dyson--Mostowski theorem. For countably generated presentations of just infinite groups with a recursively enumerable set of relations we show that in most cases the word problem is decidable. These cases include all just infinite groups which are not locally finite and (more generally) the case when there exists an algorithm enumerating an infinite set of distinct elements. Finally, we construct presentations of countably generated locally finite groups with recursively enumerable set of relations, for which the word problem is undecidable. Yet, there exist other presentations of these groups, for which the word problem is decidable.

On the word problem for just infinite groups

Abstract

We establish several results on the word problem for just infinite groups. First, for finitely generated just infinite groups we show that the word problem is uniformly decidable for presentations with recursively enumerable sets of relations. Our proof does not use the Wilson--Grigorchuk theorem on the classification of just infinite groups and proceeds directly from the definition, using ideas from classical results on decidability of the word problem: Kuznetsov's theorem and Dyson--Mostowski theorem. For countably generated presentations of just infinite groups with a recursively enumerable set of relations we show that in most cases the word problem is decidable. These cases include all just infinite groups which are not locally finite and (more generally) the case when there exists an algorithm enumerating an infinite set of distinct elements. Finally, we construct presentations of countably generated locally finite groups with recursively enumerable set of relations, for which the word problem is undecidable. Yet, there exist other presentations of these groups, for which the word problem is decidable.
Paper Structure (10 sections, 9 theorems, 13 equations)

This paper contains 10 sections, 9 theorems, 13 equations.

Table of Contents

  1. Introduction
  2. Finitely generated just infinite groups
  3. Countably generated groups
  4. Uniform word problem
  5. Word problem for a fixed presentation
  6. Monolithic groups
  7. Countable presentations with undecidable word problem
  8. Shaded sets
  9. Shaded presentations of the iterated wreath products
  10. Acknowledgments

Key Result

Theorem 1

Let $G$ be a finitely generated just infinite group. Then either $G$ is a branch group, or $G$ contains a normal subgroup of finite index isomorphic to a direct product of finitely many copies of a group $L$, where $L$ is either simple or hereditarily just infinite.

Theorems & Definitions (21)

  • Theorem
  • Theorem 1
  • proof
  • Conjecture 1
  • Theorem 2
  • proof
  • Remark 1
  • Theorem 3
  • proof
  • Theorem 4
  • ...and 11 more