The amenability problem for Thompson's group F (Geoghegan's conjecture)

Theorem 1.1: Thompson's group $F$ is not amenable. The proof finds a fixed finite set $S\subset F$ and $b>0$ with $|hA\triangle A|\ge b|A|$ for some $h\in S$, for every nonempty finite $A\subset F$, so no Folner sets exist. It converts approximate invariance of finite averages into an approximate fixed point of a Lipschitz self-map of a Hilbert ball, contradicting a Benyamini-Sternfeld displacement map that the paper constructs explicitly. It confirms Geoghegan's conjecture. The formal version gives no explicit expansion constant for a standard generating set, and the paper does not address other questions about $F$ such as its exact isoperimetric profile.

Theorem 1.1: Thompson's group $F$ is not amenable. The proof finds a fixed finite set $S\subset F$ and $b>0$ with $|hA\triangle A|\ge b|A|$ for some $h\in S$, for every nonempty finite $A\subset F$, so no Folner sets exist. It converts approximate invariance of finite averages into an approximate fixed point of a Lipschitz self-map of a Hilbert ball, contradicting a Benyamini-Sternfeld displacement map that the paper constructs explicitly. It confirms Geoghegan's conjecture. The formal version gives no explicit expansion constant for a standard generating set, and the paper does not address other questions about $F$ such as its exact isoperimetric profile.