Calculating the Fundamental Group of Galois Cover of the (2,3)-embedding of ℂℙ¹ × T

As known to all, it is quite difficult to compute the fundamental group of a surface of general type. In this paper, applying Moishezon-Teicher’s algorithm, we investigate the fundamental group of a special surface of general type with zero topological index, namely, the Galois cover of the (2,3)-embedding of ℂℙ¹ × T. Because the full presentation of the group is very complicated, we compute its special quotient and get an interesting result about its structure.