In a recent paper of Miguel N. Walsh,"Norm convergence of nilpotent ergodic averages"(http://arxiv.org/abs/1109.2922v2),the author gives a proof of the fact that multiple polynomial ergodic averages arising from nilpotent groups of measure preserving transformations of a probability space always converge in the $L^2$ norm.It is natural to ask whether the result is true for more general groups.An example is the solvable group.It is established by Milnor and Wolf in 1968 that a not virtually nilpotent solvable group has exponential growth.By a result of Bergelson and Leibman(Failure of the Roth theorem for solvable groups of exponential growth,Ergod. Th. & Dynam. Sys. (2004), 24, 45–53),in this case we can not ask for multiple convergence.