Torsion homology growth beyond asymptotics

Abstract

We show that (under mild assumptions) the generating function of log homology torsion of a knot exterior has a meromorphic continuation to the entire complex plane. As corollaries, this gives new proofs of (a) the Silver-Williams asymptotic, (b) Fried's theorem on reconstructing the Alexander polynomial (c) Gordon's theorem on periodic homology. Our results generalize to other rank 1 growth phenomena, e.g. Reidemeister-Franz torsion growth for higher-dimensional knots. We also analyze the exceptional cases where the meromorphic continuation does not exist.

Gradients of Sequences of Subgroups in a Direct Product

“…for which we assume that G is finitely presented (otherwise, (1.4) may be infinite). This gradient, having connections to number theory, geometry, and topology, became an object of intensive study, as witnessed by [2,6,8,9,10,11,12,14,24,35,36,39,42,43,49,53,54].…”