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**Groups of automorphisms of bordered orientable Klein surfaces of topological genus 2.**
*(English)*
Zbl 1383.57025

Let \(X\) be a Klein surface, that is a compact surface endowed with a dianalytic structure. This type of surface is determined topologically by three data: the topological genus \(g\), the number of boundary components \(k\) and its orientability. With these data, the algebraic genus \(p\) can be calculated. If \(p \geq 2\), the automorphism group of the surface is finite and its order is bounded above by \(12(p-1)\).

Regarding group actions, a major problem is to determine which groups act on a surface of a given topological type. Some data is known for this topic: bounds for the maximal order, groups of automorphisms of Riemann surfaces of low topological genus, groups of automorphisms of bordered Klein surfaces of low algebraic genus, groups of automorphisms of orientable Klein surfaces of topological genus 0 and 1.

In this paper, the authors obtain the groups of automorphisms of bordered orientable Klein surfaces of topological genus 2, expanding the existing abelian list for this type of surfaces. Section 2 is on the groups of automorphisms of compact orientable Klein surfaces of topological genus 2, using and expanding previous results. Also in this section, the notation is established for all the rest of the paper.

Section 3 contains the main result, in which the authors establish the admissible values of \(k\) for a given group \(G\) that is the group of automorphisms of an orientable Klein surface of topological genus 2 with \(k \geq 1\) boundary components. The proof is highly complex. The authors prove the theorem for a sample of groups that illustrate in a very clear way the procedure. The remaining groups follow by a proof similar to the one given in the paper. The proof is detailed and well explained.

In general, the paper is well written and the results are very interesting and novel.

Regarding group actions, a major problem is to determine which groups act on a surface of a given topological type. Some data is known for this topic: bounds for the maximal order, groups of automorphisms of Riemann surfaces of low topological genus, groups of automorphisms of bordered Klein surfaces of low algebraic genus, groups of automorphisms of orientable Klein surfaces of topological genus 0 and 1.

In this paper, the authors obtain the groups of automorphisms of bordered orientable Klein surfaces of topological genus 2, expanding the existing abelian list for this type of surfaces. Section 2 is on the groups of automorphisms of compact orientable Klein surfaces of topological genus 2, using and expanding previous results. Also in this section, the notation is established for all the rest of the paper.

Section 3 contains the main result, in which the authors establish the admissible values of \(k\) for a given group \(G\) that is the group of automorphisms of an orientable Klein surface of topological genus 2 with \(k \geq 1\) boundary components. The proof is highly complex. The authors prove the theorem for a sample of groups that illustrate in a very clear way the procedure. The remaining groups follow by a proof similar to the one given in the paper. The proof is detailed and well explained.

In general, the paper is well written and the results are very interesting and novel.

Reviewer: Adrián Bacelo Polo (Madrid)