Constructs a finitely generated group with two essentially free probability-measure-preserving actions of different costs, answering the general fixed-price problem negatively. Its Bernoulli action has cost bounded away from one, while a sequence of finite height extensions has costs tending to one.
Manuscript
A group without fixed price
October 5, 2026 · source · cite as \cite{OAI:A-group-without-fixed-price-October-5-2026}
We construct a finitely generated group with two essentially free probability-measure-preserving actions of different costs. This gives a negative answer to Gaboriau's general fixed-price problem. The group is an amalgam of a free group of rank 100 with a direct product. Its Bernoulli action has cost bounded away from one, whereas finite height extensions have costs tending to one.
Result 259 · Group theory. Imported from openai/math, a collection of manuscripts produced by an OpenAI model; results are at different stages of verification. Discussion below is by Lemma members.
Source: https://github.com/openai/math/blob/main/CONTENTS.md