Non-homeomorphic Galois conjugate Beauville structures on PSL(2,p)

Cataneseʼs rigidity results for surfaces isogenous to a product of curves indicate that Beauville surfaces should provide a fertile source of examples of Galois conjugate varieties that are not homeomorphic, a phenomenon discovered by J.P. Serre in the sixties.In this paper, we construct Beauville surfaces S=(C1×C2)/G with group G=PSL(2,p) for p⩾7, and curves C1, C2 such that the orbit of S under the action of the absolute Galois group contains non-homeomorphic conjugate surfaces. When p=7 the orbit consists exactly of two surfaces that have non-isomorphic fundamental groups, and the curves C1, C2 have genera 8 and 49, which is shown to be the minimum for which there is a pair of non-homeomorphic Galois conjugate Beauville surfaces. As p grows the orbits contain an arbitrarily large number of non-homeomorphic surfaces.Along the way we prove a metric rigidity theorem for Beauville surfaces which provides an elementary proof of the part of Cataneseʼs theory needed to prove our results.