Amenability of Thompson’s group F

An allowed reshaping of the interval from zero to one can compress its first half into the first quarter and stretch other pieces to keep the endpoints fixed. Thompson’s group F consists of such increasing straight-piece rules with special breakpoints and slopes. Can all its rules be averaged in a way unchanged by composing with any one of them?[1]

Can finite samples of interval-reshaping rules approximate an average that every rule preserves? One allowed rule sends the breakpoints $1/2$ and $3/4$ to $1/4$ and $1/2$, with slopes $1/2,1,2$ on its three successive pieces. Such increasing piecewise-linear maps form F; their breakpoints are dyadic and their slopes are powers of two.

Iterating just this one nonidentity rule f gives a simple example. For $A_N=\{f^{-N},\ldots,f^N\}$, right multiplication by f changes only the two end elements, so $|A_N\mathbin{\triangle}A_Nf|/|A_N|=2/(2N+1)$. This handles one cyclic subgroup, not every rule of F.

The full target is a finitely additive probability $\mu$ on every subset of F, with $\mu(F)=1$ and $\mu(Ag)=\mu(A)$ for every subset A and every $g\in F$. Finite additivity means the weights of disjoint subsets add; countable additivity is not required.[1] [2]

Researchers have proved a weaker property of representations of F and derived restrictions that any invariant average would have to obey. Neither result supplies that average or rules it out.[1] [2]

The measure paper gives conditional restrictions on possible invariant averages, not an existence proof or contradiction. The revised representation paper proves framenability for F, a weaker property, and identifies additional representation conditions relevant to amenability. The cyclic sample above does not provide finite samples that work for all generators.[1] [2]

A finite group can be averaged uniformly over all its elements, respecting its symmetries. An infinite group has no such finite list. This question connects symmetric averaging with the boundary of finite samples drawn from a concrete infinite collection of interval rules.[1] [2]

In the one-rule example, negligible boundary change explains why a finite sample can give nearly translation-invariant averages. For the whole group, the same idea must coexist with every generator. Comparing this geometry with representation conditions helps distinguish analytic descriptions that capture invariant averaging from weaker ones that F already satisfies.[1] [2]