Symmetries of hyperbolic lattices and K3 surfaces of zero entropy

Wednesday 5th February 16:00-17:00 Maths 311B

Given an even, hyperbolic lattice L, it is natural to consider

the "group of symmetries" of L, which is the quotient of the isometry

group of L by the Weyl group associated to its (-2)-roots. The isometry

group of L acts on the hyperbolic space associated to L, and the group of

symmetries can be identified with the group of isometries preserving a

certain fundamental domain. We are interested in the following two

questions:

1) When is the group of symmetries abelian or solvable, up to a finite group?

2) When is the action of the group of symmetries (on the fundamental

domain) of zero entropy (i.e., no isometry is loxodromic)?

I will show how these two questions are closely intertwined, and I will

present a geometric application to groups of automorphisms of K3 surfaces.

This is joint work with Simon Brandhorst.

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