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Let $L$ be an even, hyperbolic lattice with infinitely many simple$(-2)$-roots. We call $L$ a Borcherds lattice if it admits an isotropic vectorwith bounded inner product with all the simple $(-2)$-roots. We show that thisis the case if and only if $L$ has zero entropy, or equivalently if and only ifall symmetries of $L$ preserve some isotropic vector. We obtain a completeclassification of Borcherds lattices, consisting of $194$ lattices. In turnthis provides a classification of hyperbolic lattices of rank $\ge 5$ withvirtually solvable symmetry group. Finally, we apply these general results tothe case of K3 surfaces. We obtain a classification of Picard lattices of K3surfaces of zero entropy and infinite automorphism group, consisting of $193$lattices. In particular we show that all Kummer surfaces, all supersingular K3surfaces and all K3 surfaces covering an Enriques surface (with one exception)admit an automorphism of positive entropy.

Borcherds lattices and K3 surfaces of zero entropy