Transitive actions of finite abelian groups of sup-norm isometries

There is a long-standing conjecture of Nussbaum which asserts that every finite set in RnRn on which a cyclic group of sup-norm isometries acts transitively contains at most 2n2n 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 ΓΓ is an abelian group of sup-norm isometries that acts transitively on a finite set XX in RnRn and ΓΓ contains no anticlockwise additive chains, then XX has at most 2n2n points.