Lehmer's question, knots and surface dynamics

TL;DR

This paper links Lehmer's question to surface dynamics and knot theory, showing equivalences involving Lefschetz numbers, pseudo-Anosov homeomorphisms, and monodromies of fibered knots in lens spaces, highlighting deep connections between number theory and topology.

Contribution

It establishes new equivalences between Lehmer's question and dynamical properties of surface homeomorphisms and knot monodromies, expanding the scope of the problem.

Findings

Lehmer's question relates to growth rates of Lefschetz numbers in surface dynamics.

Equivalent formulations involve monodromies of fibered knots in lens spaces.

Connections are drawn between Perron polynomials and free group endomorphisms.

Abstract

Lehmer's question is equivalent to one about generalized growth rates of Lefschetz numbers of iterated pseudo-Anosov surface homeomorphisms. One need consider only homeomorphisms that arise as monodromies of fibered knots in lens spaces L(n,1), n>0. Lehmer's question for Perron polynomials is equivalent to one about generalized growth rates of words under injective free group endomorphisms.