Obstruction to a Higman embedding theorem for residually finite groups with solvable word problem

Emmanuel Rauzy

doi:10.1515/jgth-2020-0030

Abstract

Abstract We prove that, for a finitely generated residually finite group, having solvable word problem is not a sufficient condition to be a subgroup of a finitely presented residually finite group. The obstruction is given by a residually finite group with solvable word problem for which there is no effective method that allows, given some non-identity element, to find a morphism onto a finite group in which this element has a non-trivial image. We also prove that the depth function of this group grows faster than any recursive function.

Highlights

It is well known that there can be no Higman embedding theorem for recursively presented finitely generated residually finite groups, that is to say, not all finitely generated recursively presented residually finite groups embed into finitely presented residually finite groups
A theorem of McKinsey [3,9,11] states that all finitely presented residually finite groups have solvable word problem, while on the other hand several recursively presented residually finite groups are known that fail to have solvable word problem: for instance, one example was constructed by Meskin in [10], and one by Dyson in [4]
It was unknown whether or not the condition of having solvable word problem is sufficient for such embeddings to exist

Summary

Introduction

It is well known that there can be no Higman embedding theorem for recursively presented finitely generated residually finite groups, that is to say, not all finitely generated recursively presented residually finite groups embed into finitely presented residually finite groups. A theorem of McKinsey [3,9,11] states that all finitely presented residually finite groups have solvable word problem, while on the other hand several recursively presented residually finite groups are known that fail to have solvable word problem: for instance, one example was constructed by Meskin in [10], and one by Dyson in [4] It was unknown whether or not the condition of having solvable word problem is sufficient for such embeddings to exist. There exists a finitely generated residually finite group with solvable word problem, that does not embed in any finitely presented residually finite group. There exists a finitely generated residually finite group with solvable word problem, that is not effectively residually finite. A group has solvable word problem if and only if it is re and co-re

Dyson’s groups

Construction in Z

Non-recursive depth function