Depth of free-by-cyclic groups

TL;DR

This paper investigates invariants of free-by-cyclic groups, demonstrating that the poset of attracting lamination orbits and lamination depth are canonical and commensurability invariants, respectively, providing new tools for understanding their structure.

Contribution

It establishes that the poset of attracting lamination orbits and the lamination depth are invariants of free-by-cyclic groups, enhancing understanding of their algebraic and geometric properties.

Findings

Poset of attracting lamination orbits is a canonical invariant.

Lamination depth is a commensurability invariant.

Automorphisms with different splittings have isomorphic lamination orbit posets.

Abstract

For a free group automorphism, we prove that its poset of attracting lamination orbits is a canonical invariant of the associated mapping torus. That is, if a free-by-cyclic group splits as a mapping torus in two different ways, then the corresponding automorphisms have isomorphic posets of lamination orbits. Further, we show that the lamination depth, the size of the largest chain in this poset, is a commensurability invariant of the free-by-cyclic group.