##
**Automorphism groups of compact bordered Klein surfaces. A combinatorial approach.**
*(English)*
Zbl 0709.14021

Lecture Notes in Mathematics, 1439. Berlin etc.: Springer-Verlag. xiii, 201 p. DM 37.00 (1990).

In complex algebraic geometry, the study of the birational (or holomorphic) automorphism groups of algebraic curves (i.e., compact Riemann surfaces) has a long history, which can be traced back to the pioneering work of H. A. Schwarz, A. Hurwitz, F. Klein, P. Gordan, and A. Wiman at the end of the last century. Although the problem of determining all possible (finite) groups of automorphisms of a given Riemann surface of genus \( g\) appears very difficult, in general, and still far from being completely solved, a great progress has been achieved in the recent decades, basically due to the contributions of A. M. Macbeath, R. Accola, C. Machlachlan, H. Farkas, R. Kulkarni, and many others. The main tools for the study of automorphisms groups of Riemann surfaces are, in particular, the theory of Fuchsian groups (representing Riemann surfaces by uniformization) and the theory of Weierstrass points.

In real algebraic geometry, the study of birational automorphism groups of real curves has begun much later, actually as late as about 20 years ago. An essential step forward had been made by the paper of N. L. Alling and N. Greenleaf [cf.: “Foundations of the theory of Klein surfaces”, Lect. Notes Math. 219 (1971; Zbl 0225.30001)], in which the so-called Klein surfaces were introduced. These Klein surfaces (i.e., real surfaces admitting a dianalytic structure) turned out to be the right objects for extending the study of automorphisms of Riemann surfaces to the case of the real groundfield, in that they have real function fields in one variable as fields of “meromorphic functions” and, therefore, are categorically equivalent to real algebraic curves. On the other hand, by a fundamental result of R. Preston [cf. “Projective structures and fundamental domains on compact Klein surfaces”, Ph. D. Thesis (Univ. Texas, 1975); unpublished], there is an analogue of Poincaré’s uniformization theorem for Klein surfaces, which states that a Klein surface can be represented as a quotient of the Siegel upper-half plane by a discrete subgroup of the extended Siegel modular group. These discrete subgroups occuring here belong to the class of non-euclidean crystallographic groups (NEC groups). Finally, there is an elaborated classification theory for NEC groups, due to A. M. Macbeath [Can. J. Math. 19, 1192–1205 (1967; Zbl 0183.034)] and H. C. Wilkie [Math. Z. 91, 87–102 (1965; Zbl 0166.026)], such that the study of automorphism groups of real algebraic curves can be approached by the combinatorial study of compact bordered Klein surfaces and their representation theory via NEC groups.

This is precisely the aim of the present research monograph. The authors, who themselves contributed to the recent development of that topic by several expository papers, attempt to give a self-contained, unified and farthergoing account on the following general problem: Given a class \({\mathcal G}\) of finite groups and a class \({\mathcal K}\) of compact Klein surfaces of genus \(g\geq 2\), under which conditions do a surface S in \({\mathcal K}\) and a group G in \({\mathcal G}\) exist, such that G appears as an automorphism group of S? This problem is treated and solved in the course of the principal part of the book (chapters 3, 4, 5, 6), and that for different classes \({\mathcal G}\) of groups (e.g., cyclic groups, soluble groups, supersoluble groups, nilpotent groups, p-groups, abelian groups, etc.) and different classes \({\mathcal K}\) of Klein surfaces. Most results presented here are completely new, others are known, but scattered in the original literature.

For the convenience of the reader, the authors provide in chapter 0 an introduction to the basic notions and methods from the theories of Klein surfaces, NEC groups, and Teichmüller spaces. - In chapter 1 they give a rather comprehensive and detailed account on R. Preston’s (unpublished, cf. above) uniformization theory for Klein surfaces, which is of fundamental importance for the author’s investigations in the later chapters, whilst in chapter 2 the combinatorial presentation theory of NEC groups and their normal subgroups is given in detail. - Finally, in an appendix, a complete proof of the functorial equivalence between the category of compact bordered Klein surfaces and the category of irreducible real algebraic curves is added.

Altogether, this research monograph contains a low of new results in the hitherto fairly unexplored theory of automorphism groups of real algebraic curves. Besides that, it serves as a very useful, rather self- contained and detailed introduction to this subject of research, just as a welcome reference book on the present stage of the theory. The whole treatise is carefully and rigorously written and, as an additional feature, enhanced by a rich bibliography and (very honest and guiding) historical notes at the end of each chapter.

In real algebraic geometry, the study of birational automorphism groups of real curves has begun much later, actually as late as about 20 years ago. An essential step forward had been made by the paper of N. L. Alling and N. Greenleaf [cf.: “Foundations of the theory of Klein surfaces”, Lect. Notes Math. 219 (1971; Zbl 0225.30001)], in which the so-called Klein surfaces were introduced. These Klein surfaces (i.e., real surfaces admitting a dianalytic structure) turned out to be the right objects for extending the study of automorphisms of Riemann surfaces to the case of the real groundfield, in that they have real function fields in one variable as fields of “meromorphic functions” and, therefore, are categorically equivalent to real algebraic curves. On the other hand, by a fundamental result of R. Preston [cf. “Projective structures and fundamental domains on compact Klein surfaces”, Ph. D. Thesis (Univ. Texas, 1975); unpublished], there is an analogue of Poincaré’s uniformization theorem for Klein surfaces, which states that a Klein surface can be represented as a quotient of the Siegel upper-half plane by a discrete subgroup of the extended Siegel modular group. These discrete subgroups occuring here belong to the class of non-euclidean crystallographic groups (NEC groups). Finally, there is an elaborated classification theory for NEC groups, due to A. M. Macbeath [Can. J. Math. 19, 1192–1205 (1967; Zbl 0183.034)] and H. C. Wilkie [Math. Z. 91, 87–102 (1965; Zbl 0166.026)], such that the study of automorphism groups of real algebraic curves can be approached by the combinatorial study of compact bordered Klein surfaces and their representation theory via NEC groups.

This is precisely the aim of the present research monograph. The authors, who themselves contributed to the recent development of that topic by several expository papers, attempt to give a self-contained, unified and farthergoing account on the following general problem: Given a class \({\mathcal G}\) of finite groups and a class \({\mathcal K}\) of compact Klein surfaces of genus \(g\geq 2\), under which conditions do a surface S in \({\mathcal K}\) and a group G in \({\mathcal G}\) exist, such that G appears as an automorphism group of S? This problem is treated and solved in the course of the principal part of the book (chapters 3, 4, 5, 6), and that for different classes \({\mathcal G}\) of groups (e.g., cyclic groups, soluble groups, supersoluble groups, nilpotent groups, p-groups, abelian groups, etc.) and different classes \({\mathcal K}\) of Klein surfaces. Most results presented here are completely new, others are known, but scattered in the original literature.

For the convenience of the reader, the authors provide in chapter 0 an introduction to the basic notions and methods from the theories of Klein surfaces, NEC groups, and Teichmüller spaces. - In chapter 1 they give a rather comprehensive and detailed account on R. Preston’s (unpublished, cf. above) uniformization theory for Klein surfaces, which is of fundamental importance for the author’s investigations in the later chapters, whilst in chapter 2 the combinatorial presentation theory of NEC groups and their normal subgroups is given in detail. - Finally, in an appendix, a complete proof of the functorial equivalence between the category of compact bordered Klein surfaces and the category of irreducible real algebraic curves is added.

Altogether, this research monograph contains a low of new results in the hitherto fairly unexplored theory of automorphism groups of real algebraic curves. Besides that, it serves as a very useful, rather self- contained and detailed introduction to this subject of research, just as a welcome reference book on the present stage of the theory. The whole treatise is carefully and rigorously written and, as an additional feature, enhanced by a rich bibliography and (very honest and guiding) historical notes at the end of each chapter.

Reviewer: W.Kleinert

### MSC:

14H55 | Riemann surfaces; Weierstrass points; gap sequences |

20H15 | Other geometric groups, including crystallographic groups |

30F35 | Fuchsian groups and automorphic functions (aspects of compact Riemann surfaces and uniformization) |

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

30F50 | Klein surfaces |

14P15 | Real-analytic and semi-analytic sets |

14-02 | Research exposition (monographs, survey articles) pertaining to algebraic geometry |

30-02 | Research exposition (monographs, survey articles) pertaining to functions of a complex variable |

20-02 | Research exposition (monographs, survey articles) pertaining to group theory |