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Combinatorial group theory, Riemann surfaces and differential equations. (English) Zbl 0557.30036
Contributions to group theory, Contemp. Math. 33, 467-519 (1984).

[For the entire collection see Zbl 0539.00007.]

The authors discuss some of the interactions, resulting from the notion of the monodromy group, between group theory on the one hand and the study of Riemann surfaces and differential equations on the other hand. The paper has been written with group theory in mind and hence it contains some background material about Riemann surfaces and differential equations. The authors state that the paper is neither a survey article nor a historical account of monodromy groups, in particular, it contains new results: the introduction and application of combinatorial algorithms.

The paper provides a good introduction to the monodromy group of a branched covering and of the Hurwitz system used to obtain a cellular decomposition of it. It is described how to obtain the intersection matrix for the 1-cycles on the branched covering from the Hurwitz system. Furthermore, an algorithm is provided for calculating the intersection matrix, and it is shown how to determine a homology basis for it. There is a brief survey of results about monodromy groups of conformal self- mappings of Riemann surfaces, and a new combinatorial proof of a theorem of Hurwitz on biholomorphic self-mappings of finite order of a Riemann surface. The results are applied to determine the periods and quadratic periods of Abelian integrals on the Klein-Hurwitz curve and on the Fermat curve. Finally, there is a section on the monodromy group of a homogeneous linear differential equation.

Reviewer: V.L.Hansen

30F30Differentials on Riemann surfaces
30F10Compact Riemann surfaces; uniformization
14H30Coverings, fundamental group (curves)
57M12Special coverings
20F05Generators, relations, and presentations of groups
34C20Transformation and reduction of ODE and systems, normal forms
14C17Intersection theory, etc.