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If G has torsion we can take one of the rotation centres as base point and find a fundamental domain with 2m + 4g - 2 sides which will define m + 2g - 1 geometric generators and give the following presentation: G= (sl,... ,tg,21g I &... j=l i=l This gives an upper bound for the geometric rank, and it turns out that it is also a lower bound. 1’7. Theorem. The planar group above has geometric rank 2g + m - 1 ifm>O and2g ifm=O. Proof. The case m = 0 is trivial. Consider the Euler characteristic of lE/G, an orientable closed surface of genus g.

The oriented edge cr’ is called a neighbour of the oriented edge u if the path ~-‘a is a subpath of the boundary of a face. A boundary edge has only one neighbour and this characterizes boundary edges,except in the casewhen the edgesc, o’ have the same initial vertex which is of degree 2. A sequencegi, . . , #k of different oriented edgeswith a common initial vertex is called a star if ~j, 1 < j < k has the edgesoj-1 and cj+i as neighbours and uj-1 # aj+i. Thus at most 01 and ok can be boundary edges.

4. Example: Compact Surfaces. Let the complex S,,, (or N,,,, r > 0) have T + 1 vertices v, vi, . j~~}U{~~~,~f~:l~j~g} (or {p,“, uF1 : 1 < j < r} U {u,“’ : 1 5 j < g}, respectively,) and one pair of faces pkl with the following boundary conditions: the rj, p~j,vj start and end at v, uj runs from v to vj, ,oj from vj to vj, and dp = fiUjpjU;l . 2. 3. Examples: Closed Surfaces. 5. , rs, ~L~‘,T;‘,P~ and is closed. The Euler characteristic is x(Sg) = 2 - 29. Hence, S, and Sh with g # 1~are not homeomorphic.

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A treatise of Archimedes: Geometrical solutions derived from mechanics by Heiberg J.L. (ed.)


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