  Let $G = - \Delta_{\xi} - |\xi|^2 \frac{\partial^2}{\partial \eta^2}$ be the
Grushin operator on $\R^n \times \R.$ We prove that the Riesz transforms
associated to this operator are bounded on $L^p (\R^{n+1}), 1 < p < \infty$ and
their norms are independent of the dimension $n$.
