Axis bundles in free-by-cyclic groups

TL;DR

This paper investigates the properties of axis bundles in free-by-cyclic groups, revealing that having a unique axis is non-generic and that certain discreteness properties do not imply finiteness.

Contribution

It demonstrates the non-generic nature of monodromies with a lone axis and clarifies the limitations of discreteness implying finiteness in this context.

Findings

Lone axis property is non-generic in free-by-cyclic groups.

Associated splittings are projectively discrete in first cohomology.

Discreteness does not imply finiteness for these splittings.

Abstract

Given a splitting of a free-by-cyclic group, the associated monodromy acts on outer space preserving Handel and Mosher's "axis bundle." We show that the property of a monodromy having a "lone axis" is non-generic in the sense that the associated splittings are projectively discrete in first cohomology. Additionally, we show that this discreteness statement cannot be promoted to a finiteness statement.