Beauville surfaces, moduli spaces and finite groups
  In this paper we give the asymptotic growth of the number of connected components of the moduli space of surfaces of general type corresponding to certain families of Beauville surfaces with group either $\PSL(2,p)$, or an alternating group, or a symmetric group or an abelian group. We moreover extend these results to regular surfaces isogenous to a higher product of curves. 
Shelly Garion, Matteo Penegini
math.AG math.GR