{"id":"openai-math/A-group-without-fixed-price","pdf":"https://github.com/openai/math/blob/main/preprints/A-group-without-fixed-price-October-5-2026/unequal-costs-rank-100-amalgam.pdf","url":"https://github.com/openai/math/tree/main/preprints/A-group-without-fixed-price-October-5-2026","tags":["group-theory","ergodic-theory","orbit-equivalence"],"score":0,"title":"A group without fixed price","domain":"Group theory","family":"259","origin":"ai-generated","source":"openai-math","authors":["OpenAI"],"reviews":0,"subject":"math.GR","summary":"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.","category":"research","comments":0,"disputed":false,"division":"math","mentions":0,"discussed":0,"published":"2026-10-05","reactions":{},"formalized":false,"family_title":"A group without fixed price"}