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By Lindstrum A.

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Cm involve at least 2r+1 − (n − m) symmetries. 6, we get s1 + · · · + sm ≥ 2r+1 − n + m. We shall show that Area(Λ) ≥ 2π 2r 2r+1 − 3 − 4 |G| . Observe that each term (1/2)(1 − 1/nij ) occurring in the formula of Area(Λ) is not smaller than 1/4 because nij ≥ 2. Since G is generated by 2r+1 orientation reversing involutions, we see that |G| ≥ r+2 2 . 4 to show that Λ has a proper period ≥ |G|/2r or a link period ≥ |G|/2r+1 . In the first case s 1 + · · · + sm 2r + |G| 4 2r 2r 2r+1 − 3 2r+1 + 3n + m − 4 − > 2π − .

For the time being, this is the definition of Klein surface we shall use. The book [4] of Alling and Greenleaf and the extensive article [106] of Natanzon are excellent references for the basics on Klein surfaces. We define the genus of X as that of S. Although we shall study in detail the symmetries of the sphere and the tori, throughout this monograph, unless otherwise stated, the surfaces considered will be compact of genus ≥ 2. As said above, if Γ is an NEC group then the orbit space H/Γ admits a structure of compact Klein surface.

5). 6). We now suppose that (2s−3 − 1)/2r−s < u ≤ (2s−2 − 2)/2r−s . 7) for the second. 4). To finish the proof we consider an arbitrary integer s ∈ {4, . . , r + 2} and an arbitrary odd integer u in the range (2s−4 − 1)/2r−s < u ≤ (2s−2 − 2)/2r−s . Let G = Zs2 = x1 ⊕ · · · ⊕ xs . Let A be the set consisting of the 2s−1 involutions of G which can be written as words of odd length in x1 , . . , xs . Let us write k = 2r−s+2 u + 4 (k ≥ 5) and let Λ be a maximal NEC group with signature (0; +; [−]; {(2, .

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Abstract algebra by Lindstrum A.

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