[Paper Review] Automorphism groups of Beauville surfaces

Gareth A. Jones|arXiv (Cornell University)|Feb 15, 2011

Finite Group Theory Research12 references4 citations

TL;DR

This paper investigates the automorphism groups of Beauville surfaces of unmixed type, showing that the automorphism group $A$ is a finite solvable group with an abelian normal subgroup $I \cong Z(G)$, the center of the acting group $G$. The outer automorphism group $A/I$ embeds into the wreath product $S_3 \wr S_2$, and the authors prove that every finite abelian group arises as $I$, even when $|A:I|$ is extremal (1 or 72).

ABSTRACT

This paper shows that the automorphism group of a Beauville surface is a finite solvable group, and describes its possible structure. It relies on results of Singerman on triangle group inclusions, and of Lucchini on generators for special linear groups.

Motivation & Objective

• To characterize the structure of the full automorphism group $A = \mathrm{Aut}\,\mathcal{S}$ of a Beauville surface $\mathcal{S}$ of unmixed type.
• To determine the relationship between $A$, the inner automorphism group $I \cong Z(G)$, and the outer automorphism group $A/I$.
• To show that every finite abelian group can be realized as the inner automorphism group $I$ of some Beauville surface $\mathcal{S}$, regardless of the size of $A/I$.
• To construct examples where $A/I$ attains both the minimal value 2 and maximal value 72, achieving the full range of possible outer automorphism groups.

Proposed method

• Use results from Singerman on inclusions between hyperbolic triangle groups to bound $A/I$ as a subgroup of $S_3 \wr S_2$.
• Apply Lucchini’s results on generators of special linear groups to construct Beauville structures with prescribed center $Z(G)$.
• Construct explicit Beauville structures using finite simple groups $L_2(q)$ with triangle group quotients to realize desired automorphism groups.
• Utilize the action of field automorphisms and outer automorphisms of $G$ to induce indirect automorphisms on $\mathcal{S}$, extending $A^0$ to $A$.
• Leverage the fact that $\mathrm{Out}\,\mathcal{S} \cong A/I$ embeds into $S_3 \wr S_2$, a group of order 72, to analyze extremal cases.
• Prove that $A$ is solvable by showing $A/I$ is a subgroup of $S_3 \wr S_2$, which is solvable, and $I$ is abelian, so $A$ is solvable.

Experimental results

Research questions

• RQ1 Which finite abelian groups can arise as the inner automorphism group $I \cong Z(G)$ of a Beauville surface $\mathcal{S}$?
• RQ2 What is the possible structure of the outer automorphism group $A/I$ for a Beauville surface $\mathcal{S}$?
• RQ3 Can Beauville surfaces be constructed such that $A/I$ is as large as possible (isomorphic to $S_3 \wr S_2$) or as small as possible (trivial)?
• RQ4 Can the automorphism group $A$ of a Beauville surface be isomorphic to a generalized dihedral group?
• RQ5 How do outer automorphisms of the curves $\mathcal{C}_i$ lift to automorphisms of the surface $\mathcal{S}$?

Key findings

• The automorphism group $A$ of a Beauville surface of unmixed type is a finite solvable group.
• The inner automorphism group $I \cong Z(G)$ is an abelian normal subgroup of $A$, and every finite abelian group arises as $I$ for some Beauville surface.
• The outer automorphism group $A/I$ embeds into the wreath product $S_3 \wr S_2$, which has order 72.
• There exist Beauville surfaces for which $A/I \cong S_3 \wr S_2$, achieving the maximal possible outer automorphism group.
• There exist Beauville surfaces for which $A/I$ is trivial, achieving the minimal possible outer automorphism group.
• Every finite generalized dihedral group arises as the full automorphism group $A = \mathrm{Aut}\,\mathcal{S}$ for some Beauville surface $\mathcal{S}$.