zbMATH — the first resource for mathematics

Examples
Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

Operators
a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
Fields
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
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

MSC:
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.