Embeddings into Finitely Presented Simple Groups - Hyperbolic Groups and the Boone-Higman Conjecture

Explore a comprehensive lecture on embedding hyperbolic groups into finitely presented simple groups, delivered at the Workshop on "Geometric and Asymptotic Group Theory with Applications 2023 - Groups and Dynamics" at the Erwin Schrödinger International Institute for Mathematics and Physics. Delve into the process of embedding any hyperbolic group into a finitely presented infinite simple group, providing a proof for the "typical" case of the Boone-Higman Conjecture from 1973. Examine the talk's three main parts: a brief history of the Boone-Higman conjecture, an analysis of hyperbolic groups and their embedding into the Rational group of Grigorchuk, Nekrashevych, and Suschanskii, and a discussion on the topological full group over this rational group. Gain insights into key concepts such as the word problem for finitely generated groups, Dehn's Algorithm, small cancellation theory, and self-similar trees. Discover the collaborative work of Collin Bleak, James Belk, Francesco Matucci, and Matthew Zaremsky in this 50-minute exploration of advanced group theory concepts.

Intro
The word problem for finitely generated groups.
Key Word Problem Results
Solvable word problem for fp simple groups
The Boone-Higman Conjecture, 1974
Surface groups and Dehn's Algorithm
Dehn Presentations and Hyperbolic Groups II
Small Cancellation...
Dehn's Algorithm in Action...
The rational group
Some Terminology
Geometric Characterisation of
Hyperbolic groups are rational!
Self-Similar Trees
The Tree of Atoms
Contracting Nucleus
Then a miracle occurs...