Homology torsion growth, hyperbolic volume, and Mahler measure

Geometry Topology Seminar
Monday, November 8, 2010 - 14:00
1 hour (actually 50 minutes)
Skiles 269
We prove a conjecture of K. Schmidt in algebraic dynamical system theory onthe growth of the number of components of fixed point sets. We also prove arelated conjecture of Silver and Williams on the growth of homology torsions offinite abelian covering of link complements. In both cases, the growth isexpressed by the Mahler measure of the first non-zero Alexander polynomial ofthe corresponding modules. In the case of non-ablian covering, the growth of torsion is less thanor equal to the hyperbolic volume (or Gromov norm) of the knot complement.