Torsion numbers of augmented groups: with applications to knots and links

TL;DR

This paper explores the properties of torsion and Betti numbers associated with finitely generated groups, revealing their periodicity, recurrence relations, growth behaviors, and applications to knot and link branched cover homology.

Contribution

It introduces new results on the recurrence relations and growth rates of torsion numbers, with applications to knot and link branched cover homology.

Findings

Betti numbers are always periodic.

Torsion numbers satisfy linear recurrence relations.

p-part growth is trivial under mild conditions.

Abstract

Torsion and Betti numbers for knots are special cases of more general invariants associated to a finitely generated group G and epimorphism from G to the integers. The sequence of Betti numbers is always periodic; under mild hypotheses, the sequence of torsion numbers satisfies a linear homogeneous recurrence relation with constant coeffiencts. Generally, the torsion number sequence exhibits exponential growth rate. However, again under mild hypotheses, the p-part has trivial growth for any prime p. Applications to branched cover homology for knots and links are presented.