By Andrew O Lindstrum

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**Sample text**

Given x, y ∈ G, let (x, y) denote the union of all conjugacy classes of all powers of x, y, x y. In the unmixed case, we call G Beauville if it has two pairs of generators, {x, y} and {u, v}, such that the following holds: (∗) (x, y) ∩ (u, v) = {1} In terms of (unmixed) Beauville surfaces minimally represented as (C1 × C2 )/G where the map Ci → Ci /G = P1 (C) is ramified over 0, 1, ∞ (i = 1, 2), the fact that G is 2-generated is equivalent to it being a quotient of π1 (P1 (C) − {0, 1, ∞}) whilst (∗) is equivalent to G acting freely on C1 × C2 .

1007/978-3-319-13862-6_3 35 36 N. Boston existence of infinite paths through the tree consisting entirely or mostly of Beauville or non-Beauville groups, and the density of Beauville groups among groups of order p n for fixed n as p → ∞. 1 (Unmixed Beauville structure) Let G be a finite group. Given x, y ∈ G, let (x, y) denote the union of all conjugacy classes of all powers of x, y, x y. In the unmixed case, we call G Beauville if it has two pairs of generators, {x, y} and {u, v}, such that the following holds: (∗) (x, y) ∩ (u, v) = {1} In terms of (unmixed) Beauville surfaces minimally represented as (C1 × C2 )/G where the map Ci → Ci /G = P1 (C) is ramified over 0, 1, ∞ (i = 1, 2), the fact that G is 2-generated is equivalent to it being a quotient of π1 (P1 (C) − {0, 1, ∞}) whilst (∗) is equivalent to G acting freely on C1 × C2 .

Group Theory 13(6), 923–931 (2010) 13. B. Fairbairn, Some exceptional Beauville structures. J. Group Theory 15, 631–639 (2012) 14. Y. Fuertes, G. González-Diez, On Beauville structures on the groups Sn and An . Math. Z. 264, 959–968 (2010) 15. J. Howie, On the SQ-universality of T (6)-groups. Forum Math. 1(3), 251–272 (1989) 16. R. C. Piper, Projective Planes (Springer, New York, 1973) 17. I. Ivrissimtzis, N. Peyerimhoff, A. 2304 18. S. Immervol, A. Vdovina, Partitions of projective planes and construction of polyhedra, MaxPlanck-Institut fur Mathematik, Bonn, Preprint Series 23 (2001) 19.

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