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/pgg5f5bm1wr" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /> <![CDATA[Sp(4, 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/pgg5f5bm27s" 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/pgg5f5bm19c" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /> <![CDATA[P^4]]> </alternatives> splits as <alternatives><inline-graphic xlink:href="ark:/27927/pgg5f5bm0xs" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /> <![CDATA[Z* Z/5Z]]> </alternatives>. As a consequence, for a smooth quintic threefold <alternatives><inline-graphic xlink:href="ark:/27927/pgg5f5bm132" 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/pgg5f5bm1tn" 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/pgg5f5bm1s3" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /> <![CDATA[O_X]]> </alternatives> and by tensoring with <alternatives><inline-graphic xlink:href="ark:/27927/pgg5f5bm14m" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /> <![CDATA[O_X(1)]]> </alternatives> is an Artin group of dihedral type.