On polynomial free-by-cyclic groups

TL;DR

This paper investigates the geometric invariants of free-by-cyclic groups, focusing on the growth types of monodromies and their invariance under quasi-isometries, using cyclic and slender splittings.

Contribution

It provides a new proof that the growth type of monodromies is a geometric invariant and characterizes polynomial growth degrees via slender splittings.

Findings

Growth type of monodromy is a geometric invariant

Polynomial growth degree characterized by slender splittings

Conjecture on nesting of attracting laminations as an invariant for exponential growth

Abstract

A free-by-cyclic group can often be viewed as a mapping torus of a free group automorphism (monodromy) in multiple ways. What dynamical properties must these monodromies share, and to what extent are they invariant under quasi-isometries? We give a new proof using cyclic splittings that the growth type of a monodromy is a geometric invariant of the free-by-cyclic group; we also characterise the degree of polynomial growth using slender splittings. For exponential growth, we conjecture that the nesting of attracting laminations is a geometric invariant.