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Genus 3 mapping class groups are not Kähler
Author(s) -
Hain Richard
Publication year - 2015
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/jtu020
Subject(s) - mathematics , genus , aka , unipotent , mapping class group , class (philosophy) , bounding overwatch , combinatorics , quartic function , quadratic equation , pure mathematics , group (periodic table) , botany , geometry , computer science , surface (topology) , artificial intelligence , biology , chemistry , organic chemistry , library science
We prove that finite index subgroups of genus 3 mapping class and Torelli groups that contain the group generated by Dehn twists on bounding simple closed curves are not Kähler. These results are deduced from explicit presentations of the unipotent (aka, Malcev) completion of genus 3 Torelli groups and of the relative completions of genus 3 mapping class groups. The main results follow from the fact that these presentations are not quadratic. To complete the picture, we compute presentations of completed Torelli and mapping class in genera at least 4 ; they are quadratic. We also show that groups commensurable with hyperelliptic mapping class groups and mapping class groups in genera at most 2 are not Kähler.