The Nielsen Realization Problem for Non-Orientable Surfaces

2022/11/07 by Colin, Nestor, Xicoténcatl, Miguel A. · 1 citation
#30F10 #30F50 #30F60 #32G15 #57K20 #57M07 #Algebraic Topology (math.AT) #FOS: Mathematics

paper · doi:10.48550/arxiv.2211.03886

Abstract

We show the Teichmüller space of a non-orientable surface with marked points (considered as a Klein surface) can be identified with a subspace of the Teichmüller space of its orientable double cover. Also, it is well known that the mapping class group Mod (N_g; k) of a non-orientable surface can be identified with a subgroup of Mod (S_g-1; 2k), the mapping class group of its orientable double cover. These facts together with the classical Nielsen realization theorem are used to prove that every finite subgroup of Mod(N_g; k) can be lifted isomorphically to a subgroup of the group of diffeomorphisms Diff(N_g; k). In contrast, we show the projection Diff(N_g) → Mod(N_g) does not admit a section for large g.

Cited by

• Braid groups of the projective plane, mapping class groups of non-orientable surfaces and algebraic K-theory of their group rings

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