Let $\H$ denote the discrete Heisenberg group, equipped with a word metric
$d_W$ associated to some finite symmetric generating set. We show that if
$(X,\|\cdot\|)$ is a $p$-convex Banach space then for any Lipschitz function
$f:\H\to X$ there exist $x,y\in \H$ with $d_W(x,y)$ arbitrarily large and
\begin{equation}\label{eq:comp abs} \frac{\|f(x)-f(y)\|}{d_W(x,y)}\lesssim
\left(\frac{\log\log d_W(x,y)}{\log d_W(x,y)}\right)^{1/p}. \end{equation}

The Furstenberg recurrence theorem (or equivalently, Szemer\'edi's theorem)
can be formulated in the language of von Neumann algebras as follows: given an
integer $k \geq 2$, an abelian finite von Neumann algebra $(\M,\tau)$ with an
automorphism $\alpha: \M \to \M$, and a non-negative $a \in \M$ with
$\tau(a)>0$, one has $\liminf_{N \to \infty} \frac{1}{N} \sum_{n=1}^N \Re
\tau(a \alpha^n (a) ... \alpha^{(k-1)n} (a)) > 0$; a subsequent result of Host
and Kra shows that this limit exists. In particular, $\Re \tau(a \alpha^n (a)
>...

This is the last of three papers (following arXiv:0905.0518 and
arXiv:0910.0907) in which we develop and use some general machinery for
extending probability-preserving \bbZ^d-systems so as to obtain simplified
asymptotic behaviour for certain associated nonconventional ergodic averages.
In this third part we will use the results of the first two to obtain an
extension of an arbitrary probability-preserving \bbZ^2-system (X,\mu,T_1,T_2)
in which the associated sequences of quadratic nonconventional averages
\frac{1}{N}\sum_{n=1}^N (f