Borcherds lattices and K3 surfaces of zero entropy

Let $L$ be an even, hyperbolic lattice with infinitely many simple $(-2)$-roots. We call $L$ a Borcherds lattice if it admits an isotropic vector with bounded inner product with all the simple $(-2)$-roots. We show that this is the case if and only if $L$ has zero entropy, or equivalently if and only if all symmetries of $L$ preserve some isotropic vector. We obtain a complete classification of Borcherds lattices, consisting of $194$ lattices. In turn this provides a classification of hyperbolic lattices of rank $\ge 5$ with virtually solvable symmetry group. Finally, we apply these general results to the case of K3 surfaces. We obtain a classification of Picard lattices of K3 surfaces of zero entropy and infinite automorphism group, consisting of $193$ lattices. In particular we show that all Kummer surfaces, all supersingular K3 surfaces and all K3 surfaces covering an Enriques surface (with one exception) admit an automorphism of positive entropy.