Compression bounds for wreath products

We show that if $G$ and $H$ are finitely generated groups whose Hilbert compression exponent is positive, then so is the Hilbert compression exponent of the wreath product $G \wr H$. We also prove an analogous result for coarse embeddings of wreath products. In the special case $G=\mathbb{Z}$, $H=\mathbb{Z} \wr \mathbb{Z}$ our result implies that the Hilbert compression exponent of $\mathbb{Z}\wr (\mathbb{Z}\wr\mathbb{Z})$ is at least $1/4$, answering a question posed by several authors.