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Thin monodromy in Sp(4)

Part of: Families, fibrations

Published online by Cambridge University Press:  10 March 2014

Christopher Brav  and
Hugh Thomas
Christopher Brav
Affiliation:
Mathematical Institute, University of Oxford, 24–29 St Giles’, Oxford OX1 3LB, UK email brav@maths.ox.ac.uk
Hugh Thomas
Affiliation:
Department of Mathematics and Statistics, University of New Brunswick, Fredericton, New Brunswick, Canada E3B 5A3 email hthomas@unb.ca
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Abstract

We show that some hypergeometric monodromy groups in ${\rm Sp}(4,\mathbf{Z})$ split as free or amalgamated products and hence by cohomological considerations give examples of Zariski dense, non-arithmetic monodromy groups of real rank $2$. In particular, we show that the monodromy group of the natural quotient of the Dwork family of quintic threefolds in $\mathbf{P}^{4}$ splits as $\mathbf{Z}\ast \mathbf{Z}/5\mathbf{Z}$. As a consequence, for a smooth quintic threefold $X$ we show that the group of autoequivalences $D^{b}(X)$ generated by the spherical twist along ${\mathcal{O}}_{X}$ and by tensoring with ${\mathcal{O}}_{X}(1)$ is an Artin group of dihedral type.

Keywords

Dwork familymonodromythin groupsping-pong lemma

MSC classification

Secondary: 14D05: Structure of families (Picard-Lefschetz, monodromy, etc.)
Type
Research Article
Information
Compositio Mathematica , Volume 150 , Issue 3 , March 2014 , pp. 333 - 343
Copyright
© The Author(s) 2014 

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