  We use the Borisov-Keum equations of a fake projective plane and the
Borisov-Yeung equations of the Cartwright-Steger surface to show the existence
of a regular surface with canonical map of degree 36 and of an irregular
surface with canonical map of degree 27. As a by-product, we get equations
(over a finite field) for the $\mathbb Z/3$-invariant fibres of the Albanese
fibration of the Cartwright-Steger surface and show that they are smooth.
