By Emilio Bujalance, Jose J. Etayo, Jose M. Gamboa, Grzegorz Gromadzki
This learn monograph offers a self-contained method of the matter of opting for the stipulations less than which a compact bordered Klein floor S and a finite staff G exist, such that G acts as a bunch of automorphisms in S. The circumstances handled right here take G cyclic, abelian, nilpotent or supersoluble and S hyperelliptic or with hooked up boundary. No complicated wisdom of team idea or hyperbolic geometry is needed and 3 introductory chapters offer as a lot heritage as worthy on non-euclidean crystallographic teams. The graduate reader therefore reveals right here a simple entry to present study during this sector in addition to numerous new effects received by way of a similar unified process.
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Additional resources for Automorphism Groups of Compact Bordered Klein Surfaces: A Combinatorial Approach
1. Let f assertions are equivalent: (1) F is an NEC group. be a subgroup of an NEC group f ' . Then the following (2) The index [[":17 is finite. Proof. (1) ~ (2). It is an immediate consequence of Hurwitz Riemann formula. Now, for (2) ~ (1) notice first that F is discrete, as a subgroup of the discrete group f ' . So, all reduces to see that the topological space H/F is compact. Since fl .... ' f k E f ' H/F' is compact, F ' has a compact fundamental be the cosets representatives of f ' , that is, F ' = F f 1 U ...
2) S and S' are isomorphic if and only if F and F' are conjugate subgroups in t2=Aut(H). (3) Aut(S)~-NI2(F)/F, where NI2(/-') is the nornmlizer of F in I2. Proof. (1) Assume first that S and S' are Riemann surfaces. Let us take ~Elsom(S',S). 1. In the general case we consider the double covers f:S c >S and f . S . c. ~S' with the corresponding antianalytic involutions a:S c ----~S c and o":S~---~S c. 13, (3), there exists q/Elsom(S~,Sc) such that the square 35 commutes. S' c I// > Sc S' ~ >S Let p:H >Sc and p':H >Sc be the canonical projections.
E. the group homomorphism Mod(/-) ~Isom(T(F)):LS] z > T(fl) is not necessarily injective). 4. The following conditions are equivalent: (1) Mod(F)fails to act faithfully on T(/-'), (2) There exist an NEC group I " and a group monomorphism ot:I" that d ( / - ) = d ( / ' ) and ot(F) is a normal subgroup of F'. e. signatures of fuchsian groups) such that a=a(1-), a ' = a ( F ' ) for some groups F and F ' verifying (2) in the theorem above was obtained by Singerman in . We write [a':al=[r':a(F)].
Automorphism Groups of Compact Bordered Klein Surfaces: A Combinatorial Approach by Emilio Bujalance, Jose J. Etayo, Jose M. Gamboa, Grzegorz Gromadzki