Calibrating word problems of groups via the complexity of equivalence relations

TL;DR

This paper explores the complexity of word problems in finitely presented and finitely generated groups, revealing their deep connections to various equivalence relations and computability degrees.

Contribution

It introduces new examples of groups with highly complex and universal word problems, linking group theory with computability and equivalence relation complexity.

Findings

Existence of a finitely presented group with a uniformly effectively inseparable word problem.

Existence of a finitely generated group of computable permutations with a universal co-c.e. word problem.

Each c.e. truth-table degree contains a group's word problem.

Abstract

(1) There is a finitely presented group with a word problem which is a uniformly effectively inseparable equivalence relation. (2) There is a finitely generated group of computable permutations with a word problem which is a universal co-computably enumerable equivalence relation. (3) Each c.e. truth-table degree contains the word problem of a finitely generated group of computable permutations.