The Coxeter Quotient of the Fundamental Group of a Galois Cover of 𝕋 × 𝕋

ABSTRACT

Let X be the surface 𝕋 × 𝕋, where 𝕋 is the complex torus. This article is the third in a series studying the fundamental group of the Galois cover of X with respect to a generic projection onto ℂℙ².

Van Kampen Theorem gives a presentation of the fundamental group of the complement of the branch curve, with 54 generators and more than 2000 relations. Here we introduce a certain natural quotient (obtained by identifying pairs of generators), prove it is a quotient of a Coxeter group related to the degeneration of X, and show that this quotient is virtually nilpotent.

Communicated by C. Pedrini.

Key Words:

• Complement of branch curve
• Coxeter groups
• Fundamental group
• Galois cover

Mathematics Subject Classification:

• 14J25
• 14H30
• 20F34
• 20F55

ACKNOWLEDGMENTS

The first author was partially supported by the DAAD fellowship (Germany), Eager (Eu-network, HPRN-CT-2009-00099), and the LDFT postdoctoral fellowship (the Einstein mathematics institute, Hebrew university, Jerusalem). The Emmy Noether Research Institute for Mathematics (center of the Minerva Foundation of Germany), the Excellency Center “Group Theoretic Methods in the Study of Algebraic Varieties of the Israel Science Foundation”.