Loxodromic elements in the cyclic splitting complex and their centralizers

TL;DR

This paper characterizes when outer automorphisms act loxodromically on the cyclic splitting complex, linking this to filling laminations and cyclic splittings, and shows their centralizers are virtually cyclic.

Contribution

It provides a precise criterion for loxodromic actions on the cyclic splitting complex and establishes the virtually cyclic nature of their centralizers, extending Handel-Mosher's theorem.

Findings

Outer automorphisms with filling laminations act loxodromically

Such automorphisms have virtually cyclic centralizers

Characterization parallels Handel-Mosher's theorem for free splitting complex

Abstract

We show that an outer automorphism acts loxodromically on the cyclic splitting complex if and only if it has a filling lamination and no generic leaf of the lamination is carried by a vertex group of a cyclic splitting. This is the analog for the cyclic splitting complex of Handel-Mosher's theorem on loxodromics for the free splitting complex. We also show that such outer automorphisms have virtually cyclic centralizers.