The fundamental group of Galois covers of surfaces with octahedral envelope

Abstract

We compute the fundamental group of the Galois cover of a surface of degree~$8$, with singularities of degree $4$, whose degeneration envelope is isomorphic to an octahedron. The group is shown to be a metabelian group of order $2^{23}$. The computation amalgamates local groups, classified elsewhere, by an iterative combination of computational and group theoretic methods. Three simplified surfaces, for which the fundamental group of the Galois cover is trivial, demonstrate how nontrivial cycles in the degenerated surface complicate the computation.