Mini Workshop on Algebraic Dynamics in Saarbrücken 2025

Toward a classification of non-elementary automorphism groups of K3 surfaces up to quasi-isometry (Kohei Kikuta)
It seems hopeless to obtain a general classification of non-elementary automorphism groups of K3 surfaces up to isomorphism. Instead, I propose to study their classification up to quasi-isometry, which is weaker than isomorphism and is a standard equivalence relation in geometric group theory. Since non-elementary automorphism groups of K3 surfaces are relatively (Gromov) hyperbolic due to the speaker and Takatsu, we first focus on the classification of those automorphism groups which are hyperbolic. For Picard rank at most 3, previous results and the theory of Kleinian groups show the finiteness of quasi-isometry classes of non-elementary hyperbolic automorphism groups and provide their classification. And recently, Fujiwara-Oguiso-Yu have proved a corresponding finiteness result in the case of Picard rank at least 6. In this talk, we give an overview of the above picture, and discuss the case of Picard rank 4 via the topology of the limit sets. 

 

The Enriques surface of minimal positive entropy (Davide Veniani)
Automorphisms of algebraic surfaces can be chaotic, and their complexity is measured by a number called entropy. The smallest positive entropy possible is given by a special constant, Lehmer’s number. While Oguiso showed that no Enriques surface in characteristic 0 can achieve this, we prove in joint work with Giacomo Mezzedimi and Gebhard Martin (University of Bonn) that in characteristic 2 there is a unique Enriques surface with an automorphism whose dynamical degree equals Lehmer’s number. In this talk, I will present this result and explain the main ideas behind its proof. 

 

Calegari–Fujiwara showed that there exists a positive gap constant for the image of the stable commutator length on a hyperbolic group; this result is known as the Gap Theorem. In this talk, we introduce a norm on the automorphism group of a K3 surface arising from the entropy of its automorphisms, and we explain that an analogous Gap Theorem holds for the associated stable entropy norm. The result is based on joint work with Kohei Kikuta and Yuta Takada.