Free automorphism groups of K3 surfaces with Picard number 3

TL;DR

This paper explores the automorphism groups of certain K3 surfaces with Picard number 3, revealing they can be isomorphic to congruence subgroups of PSL_2(Z) and can include free groups of arbitrarily large rank.

Contribution

It demonstrates that specific K3 surfaces with Picard number 3 have automorphism groups isomorphic to congruence subgroups of PSL_2(Z), including free groups of any rank.

Findings

Automorphism groups are isomorphic to congruence subgroups of PSL_2(Z)

Existence of free groups of arbitrarily large rank as automorphism groups

Characterization of automorphism groups for K3 surfaces with Picard number 3

Abstract

It is known that the automorphism group of any projective K3 surface is finitely generated [24]. In this paper, we consider a certain kind of K3 surfaces with Picard number 3 whose automorphism groups are isomorphic to congruence subgroups of the modular group $PSL_2(\mathbb{Z})$. In particular, we show that a free group of arbitrarily large rank appears as the automorphism group of such a K3 surface.