By Linfan Mao

ISBN-10: 1931233926

ISBN-13: 9781931233927

A combinatorial map is a attached topological graph cellularly embedded in a floor. This monograph concentrates at the automorphism team of a map, that is relating to the automorphism teams of a Klein floor and a Smarandache manifold, additionally utilized to the enumeration of unrooted maps on orientable and non-orientable surfaces. a couple of effects for the automorphism teams of maps, Klein surfaces and Smarandache manifolds and the enumeration of unrooted maps underlying a graph on orientable and non-orientable surfaces are found. An ordinary class for the closed s-manifolds is located. Open difficulties regarding the combinatorial maps with the differential geometry, Riemann geometry and Smarandache geometries also are awarded during this monograph for the additional functions of the combinatorial maps to the classical arithmetic.

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**Extra resources for Automorphism Groups of Maps, Surfaces and Smarandache Geometries (Partially Post-Doctoral Research for the Chinese Academy of Sciences)**

**Example text**

M Particularly, since any map M can be viewed as a voltage map (M, 1G ), we get the non-Euclid area of a map M µ(M) = µ(M, 1G ) = −2πχ(M). Notice that the area of a map is only dependent on the genus of the surface. We know the following result. 5 Two maps on one surface have the same non-Euclid area. By the non-Euclid area, we get the Riemann-Hurwitz formula in Klein surface theory for a map in the following result. 1. 2, we know that |G| = = −χ(M) + −χ(M) (−1 + m∈O(F (M )) 1 ) m −2πχ(M) 2π(−χ(M) + (−1 + m∈O(F (M )) 1 )) m = µ(M) .

3, we get that |G| ≤ 4ε(M) and if G is orientation-preserving, then |G| ≤ 2ε(M). Whence, for ∀G Aut+ M, |G| ≤ 42(g ′(M) − 2) and for ∀G AutM, |G| ≤ 84(g ′(M) − 2), with the equality hold iff M is a regular map with vertex valence 3 and face valence 7 or vice via. ♮ Chapter 2 On the Automorphisms of a Klein Surface and a s-Manifold 50 Similar to the Hurwtiz theorem for a Riemann surface, we can get the upper bound for a Klein surface underlying a non-orientable surface. 5 For any Klein surface K underlying a non-orientable surface of genus q ≥ 3, |Aut+ K| ≤ 42(q − 2) and |AutK| ≤ 84(q − 2).

Hence, we can define the lifting map of a voltage map as follows. 2 For a voltage map (M, ϑ) with group G, the map M ϑ = (Xα,β , P ϑ) is called its lifting map. For a vertex v = (C)(αCα−1 ) ∈ V (M), denote by {C} the quadricells in the cycle C. The following numerical result is obvious by the definition of a lifting map. 3 Let F = (C ∗ )(αC ∗ α−1 ) be a face in the map M. Then there are |G|/o(F ) faces in the lifting map M ϑ with length |F |o(F ) lifted from the face F , where o(F ) denotes the order of ϑ(x) in the group G.

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