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Thin monodromy in Sp(4)
- Part of
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- Journal:
- Compositio Mathematica / Volume 150 / Issue 3 / March 2014
- Published online by Cambridge University Press:
- 10 March 2014, pp. 333-343
- Print publication:
- March 2014
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- Article
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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.
