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

This examine monograph presents a self-contained method of the matter of making a choice on the stipulations less than which a compact bordered Klein floor S and a finite crew G exist, such that G acts as a gaggle of automorphisms in S. The instances handled the following take G cyclic, abelian, nilpotent or supersoluble and S hyperelliptic or with attached boundary. No complex wisdom of crew concept or hyperbolic geometry is needed and 3 introductory chapters offer as a lot historical past as important on non-euclidean crystallographic teams. The graduate reader hence reveals the following a simple entry to present learn during this region in addition to numerous new effects got via an analogous unified procedure.

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**Additional resources for Automorphism Groups of Compact Bordered Klein Surfaces: A Combinatorial Approach**

**Example text**

FflN/liei x }. Consequently, U q~reih)F ' F=rE{1 ..... N//i} hE{0 ..... /i-l} is a fundamental region for F. For each r = l ..... -2 Cr: flrei (ei),flrei I (Yi0) ..... flrei 1 (Tisi),flrei 1 (7i0) ..... 1i - 2 flrei (Tis i) ..... flr(~'i0 ) ..... flr(Yis i),flr(e~) Our task now is to check that each C r generates a hole (if we identify the points in F which are equivalent under the action of F). Once this will be proved, a(/-') will consist of N/1i period-cycles P1 ..... PN/I i' are Pr: {1/2(order of the stabilizer in F of flreh(Nij))} whose elements 47 with j = 0 .....

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 .

C s belong to F. 3. Let N be even and let (n 1 ..... ns) be a nonempty period-cycle in ~r(F'), whose associated reflections are denoted {co ..... Cs}. We assume that the set J = { ( i , j ) E { 1 ..... s} x {0 ..... s-1}l i _ j , C i . l , C j + l ~ F , ci,ci+ 1 .... , c j E F } is not empty. Let us call n(i,j) the order of F(Ci_lCj+I)EF'/F. Then, for each pair (i,j)E J, we have: (1) The numbers n i and nj + 1 are even. (2) The signature of F has N/2n(i,j) copies of periods period-cycles, each consisting of n(i,j) nj + 1/2,nj,nj_ 1 .....