On zero-measured subsets of Thompson's group F

TL;DR

This paper investigates the measure-theoretic structure of Thompson's group F, showing that most parts of a natural partition have zero measure under any invariant measure, shedding light on Folner sets and the group's amenability.

Contribution

It introduces a partition of Thompson's group F via non-spherical semigroup diagrams and proves that all but one part have zero measure under any invariant measure, advancing understanding of F's structure.

Findings

All but one of the seven parts have zero measure under any right invariant finitely additive probability measure.

The results clarify the structure of Folner sets in F if the group is amenable.

Provides new insights into the measure-theoretic properties of Thompson's group F.

Abstract

A (discrete) group is called amenable whenever there exists a finitely additive right invariant probablity measure on it. For Thompson's group $F$ the problem whether it is amenable is a long-standing open question. We consider presentation of $F$ in terms of non-spherical semigroup diagrams. There is a natural partition of $F$ into 7 parts in terms of these diagrams. We show that for any measure with the above properties on $F$, all but one of these sets have zero measure. This helps to clarify the structure of Folner sets in $F$ provided the group is amenable.