Notes on automorphisms of surfaces of general type with $p_g=0$ and $K^2=7$

TL;DR

This paper studies automorphisms of certain complex surfaces with specific invariants, proving involutions are central and classifying automorphism groups of Inoue surfaces with $K^2=7$, including constructing a family with a particular automorphism group.

Contribution

It establishes that involutions are central in the automorphism group of these surfaces and classifies automorphism groups of Inoue surfaces with $K^2=7$, including new examples.

Findings

Involutions are in the center of the automorphism group.

Automorphism group of Inoue surfaces with $K^2=7$ is isomorphic to $ olinebreak bZ_2^2$ or $bZ_2 imes bZ_4$.

Constructed a 2-dimensional family of Inoue surfaces with automorphism group $bZ_2 imes bZ_4$.

Abstract

Let $S$ be a smooth minimal complex surface of general type with $p_g=0$ and $K^2=7$. We prove that any involution on $S$ is in the center of the automorphism group of $S$. As an application, we show that the automorphism group of an Inoue surface with $K^2=7$ is isomorphic to $\mathbb{Z}_2^2$ or $\mathbb{Z}_2 \times \mathbb{Z}_4$. We construct a $2$-dimensional family of Inoue surfaces with automorphism groups isomorphic to $\mathbb{Z}_2 \times \mathbb{Z}_4$.