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Artin groups and the fundamental groups of some moduli spaces
Author(s) -
Looijenga Eduard
Publication year - 2008
Publication title -
journal of topology
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 2.447
H-Index - 31
eISSN - 1753-8424
pISSN - 1753-8416
DOI - 10.1112/jtopol/jtm009
Subject(s) - mathematics , moduli space , pure mathematics , orbifold , mapping class group , coxeter group , genus , artin group , affine transformation , invertible matrix , group (periodic table) , fundamental group , moduli of algebraic curves , affine space , surface (topology) , geometry , chemistry , botany , organic chemistry , biology
We define for every affine Coxeter graph, a certain factor group of the associated Artin group and prove that some of these groups appear as orbifold fundamental groups of moduli spaces. Examples are the moduli space of nonsingular cubic algebraic surfaces and the universal nonhyperelliptic smooth genus three curve. We use this to obtain a simple presentation of the mapping class group of a compact genus three topological surface with connected boundary. This leads to a modification of Wajnryb's presentation of the mapping class groups in the higher genus case that can be understood in algebro‐geometric terms.