By Emilio Bujalance, Jose J. Etayo, Jose M. Gamboa, Grzegorz Gromadzki

This study monograph offers a self-contained method of the matter of identifying the stipulations less than which a compact bordered Klein floor S and a finite team G exist, such that G acts as a bunch of automorphisms in S. The situations handled right here take G cyclic, abelian, nilpotent or supersoluble and S hyperelliptic or with attached boundary. No complicated wisdom of team conception or hyperbolic geometry is needed and 3 introductory chapters offer as a lot heritage as precious on non-euclidean crystallographic teams. The graduate reader therefore reveals the following a simple entry to present examine during this zone in addition to numerous new effects received via a similar unified procedure.

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**Example text**

Such that 1"~1-" and d(/-')=d(F'). e. 1" is a normal subgroup of F ' . In the nonnormal case, the signatures a and a ' have genus 0 and three proper periods. 6) O" O" (0;+;[2,3,7]) (0;+;[2,3,7]) (0; + ;[2,3,7 ] ) (0;+;[2,3,8]) (0;+;[2,3,8]) (0;+;[2,3,9]) (0;+;[2,4,5]) (0; + ; [ 2 , 3 , 4 n ] ) , (0;+;[2,4,2n]), (0; + ; [ 2 , 3 , 3 n ] ) , (0;+ ;[2,3,2n]), (0;+;[7,7,7]) (0;+;[2,7,7]) (0;+;[3,3,71) (0;+;[4,8,8]) (0;+;[3,8,8]) (0;+;[9,9,9]) (0;+;[4,4,5]) (0; + ; [ n , 4 n , 4 n ] ) (0;+;[n,2n,2n]) (0;+;[3,n,3n]) (0;+;[2,n,2n]) n_>2 n>__3 n_>3 n>_4 Let A and A ' be two NEC groups with a ( A ' ) = l : ' .

I) Let e E F be an elliptic element which is not a product of two reflections in F. Then e is not a product of two reflections in F ' , because N is odd. qt -1 for some t E F ' , q E N , l_*
*

Then there exists a surface NEC group F such that S Klein surfaces. Moreover if z~':H----~H/F is the F = { f E A u t ( H ) l z c ' f = r t ' }. Notice that the conditions algebraic genus of S is bigger than or equal to 2. the algebraic genus of F. 4. The only compact topological surfaces which do not satisfy these conditions are: o r i e n t a b l e w i t h o u t boundary: g=0 g= 1 sphere t orus g=0, k=l, g=0, k=2, c losed d i s c c l o s e d annulus f g=l, k=0, p r o j e c t i v e plane t g=2, k=0, Klein bottle ~ l o r i e n t a b l e w i t h nonempty boundary: ~ l nonorientable: l g=l, k=l, M6bius s t r i p .