Enriques surfaces of zero entropy

The automorphism group of a general Enriques surface is the 2-congruence subgroup of the Weyl group of the E10-lattice. In particular, it is infinite and not virtually solvable. On the other end of the spectrum, there do exist Enriques surfaces with finite automorphism group, first classified over the complex numbers by Nikulin and Kondo. In this talk, I will explain the classification of Enriques surfaces with infinite and virtually Abelian automorphism group. This is joint work with Giacomo Mezzedimi and Davide Veniani.