Transitive actions of finite abelian groups of sup-norm isometries

There is a longstanding conjecture of Nussbaum, which asserts that every finite set in R on which a cyclic group of sup-norm isometries acts transitively contains at most 2 points. The existing evidence supporting Nussbaum’s conjecture only uses abelian properties of the group. It has therefore been suggested that Nussbaum’s conjecture might hold more generally for abelian groups of sup-norm isometries. This paper provides evidence supporting this stronger conjecture. Among other results, we show that if Γ is an abelian group of sup-norm isometries that acts transitively on a finite set X in R and Γ contains no anti-clockwise additive chains, then X has at most 2 points.