Progress around the Boone-Higman conjecture

James Belk; Collin Bleak; Francesco Matucci; Matthew C. B. Zaremsky

doi:10.4171/emss/101

Progress around the Boone-Higman conjecture

James Belk, Collin Bleak, Francesco Matucci, Matthew C. B. Zaremsky

Research output: Contribution to journal › Article › peer-review

Abstract

A conjecture of Boone and Higman from the 1970’s asserts that a finitely generated group G has solvable word problem if and only if G can be embedded into a finitely presented simple group. We comment on the history of this conjecture and survey recent results that establish the conjecture for many large classes of interesting groups.

Original language	English
Number of pages	28
Journal	EMS Surveys in Mathematical Sciences
Volume	Online First
Early online date	16 Jun 2025
DOIs	https://doi.org/10.4171/emss/101
Publication status	E-pub ahead of print - 16 Jun 2025

Keywords

Boone-Higman conjecture
Thompson group
Finitely presented
Simple group
Word problem

Access to Document

10.4171/emss/101

https://arxiv.org/abs/2306.16356v3

1 Preprint

Progress around the Boone-Higman conjecture
Belk, J., Bleak, C., Matucci, F. & Zaremsky, M. C. B., 28 Jun 2023, arXiv, 21 p.
Research output: Working paper › Preprint

Cite this

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title = "Progress around the Boone-Higman conjecture",

abstract = "A conjecture of Boone and Higman from the 1970{\textquoteright}s asserts that a finitely generated group G has solvable word problem if and only if G can be embedded into a finitely presented simple group. We comment on the history of this conjecture and survey recent results that establish the conjecture for many large classes of interesting groups.",

keywords = "Boone-Higman conjecture, Thompson group, Finitely presented, Simple group, Word problem",

author = "James Belk and Collin Bleak and Francesco Matucci and Zaremsky, \{Matthew C. B.\}",

note = "Funding: The third author gratefully acknowledges the support of the Fundacao para a Ciencia e a Tecnologia (CEMAT-Ciencias FCT projects UIDB/04621/2020 and UIDP/04621/2020) and of the Universita degli Studi di Milano–Bicocca (FA project 2020-ATE-0006 “Strutture Algebriche”). The fourth author is supported by grant \#635763 from the Simons Foundation.",

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AU - Matucci, Francesco

AU - Zaremsky, Matthew C. B.

N1 - Funding: The third author gratefully acknowledges the support of the Fundacao para a Ciencia e a Tecnologia (CEMAT-Ciencias FCT projects UIDB/04621/2020 and UIDP/04621/2020) and of the Universita degli Studi di Milano–Bicocca (FA project 2020-ATE-0006 “Strutture Algebriche”). The fourth author is supported by grant #635763 from the Simons Foundation.

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N2 - A conjecture of Boone and Higman from the 1970’s asserts that a finitely generated group G has solvable word problem if and only if G can be embedded into a finitely presented simple group. We comment on the history of this conjecture and survey recent results that establish the conjecture for many large classes of interesting groups.

AB - A conjecture of Boone and Higman from the 1970’s asserts that a finitely generated group G has solvable word problem if and only if G can be embedded into a finitely presented simple group. We comment on the history of this conjecture and survey recent results that establish the conjecture for many large classes of interesting groups.

KW - Boone-Higman conjecture

KW - Thompson group

KW - Finitely presented

KW - Simple group

KW - Word problem

U2 - 10.4171/emss/101

DO - 10.4171/emss/101

M3 - Article

SN - 2308-2151

VL - Online First

JO - EMS Surveys in Mathematical Sciences

JF - EMS Surveys in Mathematical Sciences

ER -

Progress around the Boone-Higman conjecture

Abstract

Keywords

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Research output

Progress around the Boone-Higman conjecture

Cite this