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.
So we are left with the virtually nilpotent case,i.e.we want to know that when $T_1,T_2,...T_l$ are measure preserving transformations of a probability Space $(X,\chi,\mu)$ which generate a virtually nilpotent group,whether the averages$\frac{1}{N}\sum_{n=1}^N\Pi_{i=1}^{d}(T_1^{p_{1,j}(n)}T_2^{p_{2,j}(n)}...T_l^{p_{l,j}(n)})f_j\ $converges in $L^2(X,\chi,\mu)$,for every $f_1,...,f_d\in L^{\infty}(X,\chi,\mu)$ and every set of integer valued polynomials $p_{i,j}$.
Having Walsh's result in hand,I wonder whether there is a way to deduce the virtual nilpotent group case from the nilpotent case,since virtual nipotent is not far from nilpotent(for this reason,I think the result is more likely to be true),or generalize Walsh's definition of complexity and educible functions so that we can use it to get virtual nilpotent case.
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