The bicanonical map of the Cartwright-Steger surface

We prove that the bicanonical map of the Cartwright-Steger surface is an embedding. We also discuss two minimal surfaces of general type, both covered by the Cartwright-Steger surface. One has $K^2=2$, $p_g=1$, ${\pi }_1=\{1\}$ and the other has $K^2=1$, $p_g=0$, ${\pi }_1=\mathbb{Z}\mathrm{/}2\mathbb{Z}$.