##
**Finite group actions on bordered surfaces of small genus.**
*(English)*
Zbl 1209.30016

A compact bordered Klein surface may be seen as the quotient of an unbordered orientable surface under the action of a symmetry with fixed points. A bordered surface may be endowed with a dianalytic structure and is called a Klein surface. Most Klein surfaces admit only the trivial automorphism. Surfaces with algebraic genus \(p>2\) with a non-trivial automorphism constitute the singular locus of the moduli space of compact Klein surfaces of a given topological type.

Many results about the group of automorphisms of bordered Klein surfaces are known, but the list of all finite groups of automorphisms of these surfaces are known only for algebraic genus \(2\) and \(3\), and they were obtained in [E. Bujalance, J. M. Gamboa, Arch. Math. 42, 229–237 (1984; Zbl 0522.14016)] and [E. Bujalance, J. J. Etayo, J. M. Gamboa, Proc. Japan Acad. Ser. A. 62, 40–42 (1986; Zbl 0594.14019)], respectively.

A very interesting problem is to determine for each group of automorphisms of a bordered surface the inequivalent topological group actions. This is a finer classification and it is necessary for the study of the singular locus in the moduli space. This space can be viewed as the quotient of the Teichmüller space under the action of the mapping class group. The conjugacy classes of finite subgroups of the mapping class group are in one-to-one correspondence with the topological classes of finite groups actions.

In the paper under review, the authors determine, up to topological equivalence, all finite group actions on compact bordered surfaces of algebraic genus \(p\) for \(2\leq p\leq 6\). These surfaces can be orientable or non-orientable, and they are topologically classified by means of three parameters, namely, the algebraic genus \(p\), the number of boundary components \(k\), and its orientability. The triple \((p,k,\epsilon)\), where \(\epsilon=+\) or \(\epsilon=-\) according to the surface is orientable or not, is called the topological type of the surface. By a classical result of Weichold, for a fixed value \(p\), there are \([(3p+4)/2]-1]\) different types of bordered surfaces, where \([t]\) denotes the integral part of \(t\). In the same way as Fuchsian groups play an important role in the study of unbordered orientable surfaces, \(NEC\) groups are an important tool for Klein surfaces.

The authors deal first with groups of order up to \(5\). They find all the topological types of bordered surfaces of any genus on which these groups act. In addition, groups of prime odd order are also included. Then they study the outer automorphism groups of certain \(NEC\) groups and give an explicit description of the effect of certain outer automorphisms on the generators of the \(NEC\) group. In the last section of the paper, they apply the obtained results to determine the inequivalent actions of groups of order at least \(6\) on compact bordered Klein surfaces of algebraic genus \(p=2,3,4,5,6\). All actions are obtained with the help of an algorithm for finding all normal subgroups of given small index in a finitely presented group and by means of the algebra system MAGMA.

Then the authors deal with groups of order at least \(6\). For algebraic genera \(p=2, 3, 4\), all inequivalent actions for each topological type \((p, k, \epsilon)\) are explicitly given in tables displaying each group \(G\), a presentation for \(G\), the branch data, and the generator vector. It follows that there are \(10, 37\), and \(81\) inequivalent actions on bordered surfaces if \(p\) is \(2, 3\), or \(4\), respectively. Moreover, there does not exist any group of order \(6\) or more acting on a bordered surface of topological type \((2,1,-)\).

When \(p=5,6\), the numbers of inequivalent actions are \(273\) and \(216\), respectively, and so the tables would be too large, and they are not included in the paper. Instead, two tables are provided giving explicitly the number of inequivalent actions of each group on each topological type. From these tables we can see for example that the type \((5,2,+)\) admits \(64\) different actions, that the group \(D_4\) gives \(71\) different actions on surfaces of algebraic genus \(5\), and that the group \(C_2\times C_2\times C_2\) can act on surfaces of topological type \((5,2,-)\) in \(24\) different ways.

There is a slight mistake in Remark 3.3. The authors claim that \(C_2\times C_2\) does not act on surfaces of type \((p,0,-)\) (unbordered non-orientable surfaces) when \(p\) is even, which is wrong, but it does not affect the results of the paper.

In the opinion of the reviewer, it is a very valuable work, well written and with a nice and motivating general introduction. The used techniques are well described. In the last section, about the actions of groups of order at least \(6\), there is an unusual long introduction to the section carefully describing the used algorithm. The authors also give many clarifying examples throughout the paper.

Many results about the group of automorphisms of bordered Klein surfaces are known, but the list of all finite groups of automorphisms of these surfaces are known only for algebraic genus \(2\) and \(3\), and they were obtained in [E. Bujalance, J. M. Gamboa, Arch. Math. 42, 229–237 (1984; Zbl 0522.14016)] and [E. Bujalance, J. J. Etayo, J. M. Gamboa, Proc. Japan Acad. Ser. A. 62, 40–42 (1986; Zbl 0594.14019)], respectively.

A very interesting problem is to determine for each group of automorphisms of a bordered surface the inequivalent topological group actions. This is a finer classification and it is necessary for the study of the singular locus in the moduli space. This space can be viewed as the quotient of the Teichmüller space under the action of the mapping class group. The conjugacy classes of finite subgroups of the mapping class group are in one-to-one correspondence with the topological classes of finite groups actions.

In the paper under review, the authors determine, up to topological equivalence, all finite group actions on compact bordered surfaces of algebraic genus \(p\) for \(2\leq p\leq 6\). These surfaces can be orientable or non-orientable, and they are topologically classified by means of three parameters, namely, the algebraic genus \(p\), the number of boundary components \(k\), and its orientability. The triple \((p,k,\epsilon)\), where \(\epsilon=+\) or \(\epsilon=-\) according to the surface is orientable or not, is called the topological type of the surface. By a classical result of Weichold, for a fixed value \(p\), there are \([(3p+4)/2]-1]\) different types of bordered surfaces, where \([t]\) denotes the integral part of \(t\). In the same way as Fuchsian groups play an important role in the study of unbordered orientable surfaces, \(NEC\) groups are an important tool for Klein surfaces.

The authors deal first with groups of order up to \(5\). They find all the topological types of bordered surfaces of any genus on which these groups act. In addition, groups of prime odd order are also included. Then they study the outer automorphism groups of certain \(NEC\) groups and give an explicit description of the effect of certain outer automorphisms on the generators of the \(NEC\) group. In the last section of the paper, they apply the obtained results to determine the inequivalent actions of groups of order at least \(6\) on compact bordered Klein surfaces of algebraic genus \(p=2,3,4,5,6\). All actions are obtained with the help of an algorithm for finding all normal subgroups of given small index in a finitely presented group and by means of the algebra system MAGMA.

Then the authors deal with groups of order at least \(6\). For algebraic genera \(p=2, 3, 4\), all inequivalent actions for each topological type \((p, k, \epsilon)\) are explicitly given in tables displaying each group \(G\), a presentation for \(G\), the branch data, and the generator vector. It follows that there are \(10, 37\), and \(81\) inequivalent actions on bordered surfaces if \(p\) is \(2, 3\), or \(4\), respectively. Moreover, there does not exist any group of order \(6\) or more acting on a bordered surface of topological type \((2,1,-)\).

When \(p=5,6\), the numbers of inequivalent actions are \(273\) and \(216\), respectively, and so the tables would be too large, and they are not included in the paper. Instead, two tables are provided giving explicitly the number of inequivalent actions of each group on each topological type. From these tables we can see for example that the type \((5,2,+)\) admits \(64\) different actions, that the group \(D_4\) gives \(71\) different actions on surfaces of algebraic genus \(5\), and that the group \(C_2\times C_2\times C_2\) can act on surfaces of topological type \((5,2,-)\) in \(24\) different ways.

There is a slight mistake in Remark 3.3. The authors claim that \(C_2\times C_2\) does not act on surfaces of type \((p,0,-)\) (unbordered non-orientable surfaces) when \(p\) is even, which is wrong, but it does not affect the results of the paper.

In the opinion of the reviewer, it is a very valuable work, well written and with a nice and motivating general introduction. The used techniques are well described. In the last section, about the actions of groups of order at least \(6\), there is an unusual long introduction to the section carefully describing the used algorithm. The authors also give many clarifying examples throughout the paper.

Reviewer: Ernesto Martínez (Madrid)

### MSC:

30F50 | Klein surfaces |

57M60 | Group actions on manifolds and cell complexes in low dimensions |

20H10 | Fuchsian groups and their generalizations (group-theoretic aspects) |

### Software:

Magma
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XMLCite

\textit{E. Bujalance} et al., J. Pure Appl. Algebra 214, No. 12, 2165--2185 (2010; Zbl 1209.30016)

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### References:

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