Groups with decidable word problem that do not embed in groups with decidable conjugacy problem

TL;DR

This paper constructs finitely presented torsion-free groups with decidable word problems that cannot embed into any finitely generated group with decidable conjugacy problem, answering a longstanding open question.

Contribution

It provides the first example of such groups, demonstrating a separation between decidability of word and conjugacy problems in group embeddings.

Findings

Existence of finitely presented torsion-free groups with decidable word problem

These groups cannot embed into any finitely generated group with decidable conjugacy problem

Answers a question posed by Collins in the 1970s

Abstract

We show the existence of finitely presented torsion-free groups with decidable word problem that cannot be embedded in any finitely generated group with decidable conjugacy problem. This answers a well-known question of Collins from the early 1970's.