Symmetry groups and fundamental tilings for the compact surface of genus $3^-$. 2. The normalizer diagram with classification

Emil Molnár and Eleonóra Stettner

Abstract: This is a continuation of [S1] where the complete diagram of metric normalizers of the fundamental group {\bf G}$\otimes^3$ in Isom$\;H^2$ will be determined (Table 2). Thus we completely classify the symmetry groups {\bf N}/{\bf G} of the $3^-$ surface, i.e. the connected sum of 3 projective planes, into 12 normalizer classes, up to topological equivariance, by the algorithm for fundamental domains, developed in [LM1], [LM2], [LM3] and [S2], aided by computer. Our algorithm is applicable for any compact surface with exponential complexity by the genus $g$.

[S1] Stettner, E.: Symmetriegruppen und fundamentale Pflasterungen der Fläche vom Geschlecht -3. I. Maximale Gruppen mit Sechseckbereichen. Studia Sci. Math. Hung. {\bf 40} (2003), 41--57.

[LM1] Lu{\v c}i{\'c}, Z.; Moln{á}r, E.: Combinatorial classification of fundamental domains of finite area for planar discontinuous isometry groups. Arch. Math. {\bf 54} (1990), 511--520.

[LM2] Lu{\v c}i{\'c}, Z.; Moln{á}r, E.: Fundamental domains for planar discontinuous groups and uniform tilings. Geom. Dedicata {\bf 40} (1991), 125--143.

[LM3] Lu{\v c}i{\'c}, Z.; Moln{á}r, E.; Vasiljevi{\'c}, N.: Combinatorial structure of fundamental polygons of finite area for plane discontinuous groups. Manuscript 1998.

[S2] Stettner, E.: Die Computergestützte Klassifizierung der Flächeneinwickelungen in einem Vieleck vorgegebener Seitenanzahl. Annales Univ. Sci. Budapest {\bf 41} (1998), 103--115.

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