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Two results on complexities of decision problems of groups

Uri Andrews, Matthew Harrison-Trainor, Meng-Che "Turbo" Ho

Abstract

We answer two questions on the complexities of decision problems of groups, each related to a classical result. First, C. Miller characterized the complexity of the isomorphism problem for finitely presented groups in 1971. We do the same for the isomorphism problem for recursively presented groups. Second, the fact that every Turing degree appears as the degree of the word problem of a finitely presented group is shown independently by multiple people in the 1960s. We answer the analogous question for degrees of ceers instead of Turing degrees. We show that the set of ceers which are computably equivalent to a finitely presented group is $Σ^0_3$-complete, which is the maximal possible complexity.

Two results on complexities of decision problems of groups

Abstract

We answer two questions on the complexities of decision problems of groups, each related to a classical result. First, C. Miller characterized the complexity of the isomorphism problem for finitely presented groups in 1971. We do the same for the isomorphism problem for recursively presented groups. Second, the fact that every Turing degree appears as the degree of the word problem of a finitely presented group is shown independently by multiple people in the 1960s. We answer the analogous question for degrees of ceers instead of Turing degrees. We show that the set of ceers which are computably equivalent to a finitely presented group is -complete, which is the maximal possible complexity.
Paper Structure (5 sections, 13 theorems, 4 equations, 1 figure)

This paper contains 5 sections, 13 theorems, 4 equations, 1 figure.

Table of Contents

  1. Introduction
  2. The Isomorphism Problem
  3. The Word Problem
  4. The isomorphism problem for finitely generated groups
  5. Classifying the ceers equivalent to a word problem of groups

Key Result

Theorem 1.1

The isomorphism problem for $6$-generated recursively presented groups is a $\Sigma^0_3$-complete equivalence relation, i.e., every other $\Sigma^0_3$ equivalence relation computably reduces to it.

Figures (1)

  • Figure 1: An example where $j-i \notin V_r$ but $j'-i\in V_r$.

Theorems & Definitions (34)

  • Theorem 1.1
  • Definition 1.2
  • Theorem 1.3
  • Definition 2.1
  • Theorem 2.2: An application of Andrews and San Mauro ASM1
  • Lemma 2.3
  • proof
  • Theorem 2.4
  • proof
  • Claim 2.4.1
  • ...and 24 more