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International Mathematics Research Notices Advance Access published online on January 21, 2009
International Mathematics Research Notices, doi:10.1093/imrn/rnn164
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© The Author 2009. Published by Oxford University Press. All rights reserved. For permissions, please e-mail: journals.permissions@oxfordjournals.org

Lefschetz Fibrations and an Invariant of Finitely Presented Groups

Mustafa Korkmaz

Department of Mathematics, Middle East Technical University, 06531 Ankara, Turkey

Correspondence: Correspondence to be sent to: korkmaz{at}metu.edu.tr

Every finitely presented group is the fundamental group of the total space of a Lefschetz fibration. This follows from results of Gompf and Donaldson, and was also proved by Amoros–Bogomolov–Katzarkov–Pantev. We give another proof by providing the monodromy explicitly. We then define the genus of a finitely presented group {Gamma} to be the minimal genus of a Lefschetz fibration with fundamental group {Gamma}. We also give some estimates of the genus of certain groups.


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