Abelian actions on compact nonorientable Riemann surfaces. (English) Zbl 1497.14049
Let \(X\) be a non-orientable Riemann surface of topological genus \(g > 2\). Then, its automorphism group is finite. If \(G\) is a subgroup of the automorphism group of \(X\), then there exist a non-Euclidean crystallographic (NEC) group \(\Lambda\), and an epimorphism \(\theta : \Lambda \rightarrow G\) such that \(\mathcal{H} / \ker\theta \approx X\), where \(\mathcal{H}\) is the hyperbolic plane.
In the paper under review, the author considers \(G\) an abelian group. The main result is Theorem 4.3: Given an NEC group \(\Lambda\) and an abelian finite group \(A\), the result provides the necessary and sufficient conditions for the existence of the above epimorphism \(\theta : \Lambda \rightarrow A\). These conditions are expressed in terms of the signature of the NEC group \(\Lambda\) and of the invariant factors of \(A\). It is worth to note that the proof of this result is really complicated, but very well and clearly explained.
As a byproduct of this result, he reobtains in a new way the symmetric crosscap number of \(A\), \(\tilde {\sigma} (A)\), which is the least genus of the non-orientable Riemann surfaces on which \(A\) acts as a subgroup of the automorphism group. The value of \(\tilde {\sigma} (A)\) was originally obtained by E. Bujalance in [Pac. J. Math. 109, 279–289 (1983; Zbl 0545.30033)] for cyclic groups; by the reviewer in [Sobre grupos de automorfismos de superficies de Klein. Univ. Complutense (Doctoral Thesis) (1983)] for non-cyclic abelian groups of odd order; and by G. Gromadzki in [Ann. Soc. Math. Pol, Ser. I, Commentat. Math. 28, No. 2, 197-217 (1989; Zbl 0693.30038)] for all abelian groups.
By using Theorem 4.3, the author obtains the least symmetric crosscap number of the abelian groups of a given order, as well as the largest order of an abelian group acting on a non-orientable Riemann surface of genus \(g > 2\). This result on the maximum order problem expands those obtained for cyclic groups by Bujalance in [loc. cit.] and by the reviewer in [loc. cit] for non-cyclic abelian groups of odd order.
In the paper under review, the author considers \(G\) an abelian group. The main result is Theorem 4.3: Given an NEC group \(\Lambda\) and an abelian finite group \(A\), the result provides the necessary and sufficient conditions for the existence of the above epimorphism \(\theta : \Lambda \rightarrow A\). These conditions are expressed in terms of the signature of the NEC group \(\Lambda\) and of the invariant factors of \(A\). It is worth to note that the proof of this result is really complicated, but very well and clearly explained.
As a byproduct of this result, he reobtains in a new way the symmetric crosscap number of \(A\), \(\tilde {\sigma} (A)\), which is the least genus of the non-orientable Riemann surfaces on which \(A\) acts as a subgroup of the automorphism group. The value of \(\tilde {\sigma} (A)\) was originally obtained by E. Bujalance in [Pac. J. Math. 109, 279–289 (1983; Zbl 0545.30033)] for cyclic groups; by the reviewer in [Sobre grupos de automorfismos de superficies de Klein. Univ. Complutense (Doctoral Thesis) (1983)] for non-cyclic abelian groups of odd order; and by G. Gromadzki in [Ann. Soc. Math. Pol, Ser. I, Commentat. Math. 28, No. 2, 197-217 (1989; Zbl 0693.30038)] for all abelian groups.
By using Theorem 4.3, the author obtains the least symmetric crosscap number of the abelian groups of a given order, as well as the largest order of an abelian group acting on a non-orientable Riemann surface of genus \(g > 2\). This result on the maximum order problem expands those obtained for cyclic groups by Bujalance in [loc. cit.] and by the reviewer in [loc. cit] for non-cyclic abelian groups of odd order.
Reviewer: José Javier Etayo (Madrid)
MSC:
| 14H15 | Families, moduli of curves (analytic) |
| 57M60 | Group actions on manifolds and cell complexes in low dimensions |
| 20F05 | Generators, relations, and presentations of groups |
| 20H10 | Fuchsian groups and their generalizations (group-theoretic aspects) |
| 30F50 | Klein surfaces |
| 14H37 | Automorphisms of curves |
References:
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