Consider a surface of general type with canonical map of finite degree $d$, and with irregularity $q$.

It is known since the seminal paper of Beauville (1979) that either $q=0, d\leq 36$ or $q>0, d\leq 27$.

So far only examples with degree $d\leq 32$ have been constructed.

{\em Ball quotient surfaces} are usually very hard to construct explicitly and deal with. Recently two breakthrough papers appeared, Borisov-Keum and Borisov-Yeung,

where two certain such surfaces are given by equations, namely one of the 100 {\em fake projective planes} and the so-called {\em Cartwright-Steger surface}.

In this talk I will prove the existence of the above boundary cases $q=0, d=36$ and $q>0, d=27$.

The hard part of the proof consists on finding equations for certain algebraic curves in that two ball quotient surfaces. All computations are implemented with the computer algebra system Magma.