Failure of the Roth theorem for solvable groups of exponential growth

Abstract

We show that for any finitely generated solvable group of exponential growth one can find a measure-preserving action for which the multiple recurrence theorem fails and a measure-preserving action for which the ergodic Roth theorem fails. This is in contrast with the positive results established by Leibman (Geom. Funct. Anal.8 (1998), 853–931) and Bergelson and Leibman (Invent. Math.147 (2002), 429–470) for nilpotent group actions.