Thin monodromy in Sp(4). (March 2014) - Brav, Christopher; Thomas, Hugh
Abstract We show that some hypergeometric monodromy groups in <alternatives><inline-graphic xlink:href="ark:/27927/pghjmhh66k" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /> <![CDATA[${\rm Sp}(4, \mathbf{Z})$]]> </alternatives> split as free or amalgamated products and hence by cohomological considerations give examples of Zariski dense, non-arithmetic monodromy groups of real rank <alternatives><inline-graphic xlink:href="ark:/27927/pghjmhh6xf" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /> <![CDATA[$2$]]> </alternatives>. In particular, we show that the monodromy group of the natural quotient of the Dwork family of quintic threefolds in <alternatives><inline-graphic xlink:href="ark:/27927/pghjmhh4sd" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /> <![CDATA[$\mathbf{P}^{4}$]]> </alternatives> splits as <alternatives><inline-graphic xlink:href="ark:/27927/pghjmhh3rd" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /> <![CDATA[$\mathbf{Z}\ast \mathbf{Z}/5\mathbf{Z}$]]> </alternatives>. As a consequence, for a smooth quintic threefold <alternatives><inline-graphic xlink:href="ark:/27927/pghjmhh432" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /> <![CDATA[$X$]]> </alternatives> we show that the group of autoequivalences <alternatives><inline-graphic xlink:href="ark:/27927/pghjmhh610" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /> <![CDATA[$D^{b}(X)$]]> </alternatives> generated by the spherical twist along <alternatives><inline-graphic xlink:href="ark:/27927/pghjmhh5wf" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /> <![CDATA[${\mathcal{O}}_{X}$]]> </alternatives> and by tensoring with <alternatives><inline-graphic xlink:href="ark:/27927/pghjmhh474" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /> <![CDATA[${\mathcal{O}}_{X}(1)$]]> </alternatives> is an Artin group of dihedral type.