On the virtual cohomological dimensions of automorphism groups of K3 surfaces

We will discuss Mukai’s conjecture that the virtual cohomological dimension of the automorphism group of a K3 surface is equal to the maximum rank of its Mordell-Weil groups. The action of the automorphism group on the second cohomology induces a natural action on a hyperbolic space. In this talk, we will explain that Mukai’s conjecture can be regarded as a problem in hyperbolic geometry and geometric group theory via this action. In particular, we give the formula that determines the virtual cohomological dimension of the automorphism group of a K3 surface by the covering dimension of the blown-up boundary associated with the ample cone of the K3 surface. If time permits, we will give an affirmative example and a counterexample of Mukai’s conjecture.