Swiss Knots 2011

Knot Theory and Algebra

Lake Thun, May 23-27

Torsion growth and Mahler measure

Prof. Dr. Thang Le's talk
Date: 24.05.11   Time: 10.00 - 10.50   Room: Gwatt Zentrum

Abstract: We establish a conjecture of K. Schmidt in algebraic dynamical system theory on the growth of the number of components of fixed point sets. We also prove a related conjecture of Silver and Williams on the growth of homology torsions of finite abelian covering of link complements. In both cases, the growth is expressed by the Mahler measure of the first non-zero Alexander polynomial of the corresponding modules. We use the theory of pseudo-isomorphisms in commutative algebra and the theory of torsion points on algebraic varieties. We also describe concrete sequences which give the expected values of the limits in both conjectures. For this part we utilize a result of Bombieri and Zannier (conjectured before by A. Schinzel) and a result of Lawton (conjectured before by D. Boyd).