Monodromy representations and surfaces with maximal Albanese dimension

We relate the existence of some surfaces of general type and maximal Albanese dimension to the existence of some monodromy representations of the braid group B2(C2) in the symmetric group Sn. Furthermore, we compute the number of such representations up to n= 9 , and we analyze the cases n∈{2,3,4}. For n=2,3 we recover some surfaces with pg= q= 2 recently studied (with different methods) by the author and his collaborators, whereas for n= 4 we obtain some conjecturally new examples.