Thin monodromy in Sp(4)

TL;DR

This paper investigates the structure of certain hypergeometric monodromy groups in Sp(4,Z), revealing their splitting properties and implications for non-arithmetic monodromy groups and derived autoequivalence groups of quintic threefolds.

Contribution

It demonstrates that some hypergeometric monodromy groups in Sp(4,Z) split as free or amalgamated products, providing new examples of non-arithmetic, Zariski dense monodromy groups of real rank 2.

Findings

Hypergeometric monodromy groups in Sp(4,Z) can split as free or amalgamated products.

The monodromy of the Dwork family quotient splits as Z*Z/5.

Autoequivalence groups of smooth quintic threefolds relate to Artin groups of dihedral type.

Abstract

We show that some hypergeometric monodromy groups in Sp(4,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 of the natural quotient of the Dwork family of quintic threefolds in P^{4} splits as Z*Z/5. As a consequence, for a smooth quintic threefold X we show that a certain group of autoequivalences of the bounded derived category of coherent sheaves is an Artin group of dihedral type.