Perspectives on V

Richard Garner – 1 February 2023

In the 1970's, Graham Higman gave a description of Thompson's group V as the automorphism group of the free Jonsson--Tarski algebra with one generator, where a Jonsson--Tarski algebra is a set X endowed with an isomorphism to X * X.

Higman's original approach was to calculate this free algebra explicitly, and via some combinatorics present its automorphism group V in a more tractable way. Subsequently, it has become popular to describe V in terms of certain maps between ideals of the free monoid on two generators {l,r}, and this requires further combinatorics to equate with Higman's description. In this talk, we explain why the combinatorics are unneccessary: we can get the ideal-theoretic description almost trivially from Freyd's description of the category of Jonsson--Tarski algebras as a category of sheaves.