Compression bounds for wreath products

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