The geometry of generalized loxodromic elements

TL;DR

This paper investigates geometric criteria that determine when elements in finitely generated groups are generalized loxodromic, extending existing results and providing sharp conditions for small cancellation groups.

Contribution

It generalizes Sisto's result that all generalized loxodromic elements are Morse and offers new geometric conditions for identifying such elements in small cancellation groups.

Findings

Generalized loxodromic elements are Morse.

Provided geometric conditions for small cancellation groups.

Constructed examples demonstrating sharpness of conditions.

Abstract

We explore geometric conditions which ensure a given element of a finitely generated group is, or fails to be, generalized loxodromic; as part of this we prove a generalization of Sisto's result that every generalized loxodromic element is Morse. We provide a sufficient geometric condition for an element of a small cancellation group to be generalized loxodromic in terms of the defining relations and provide a number of constructions which prove that this condition is sharp.