A non-loxodromic Morse element in a Morse local-to-global group

TL;DR

This paper constructs a Morse local-to-global group containing an infinite-order Morse element that is not loxodromic in any hyperbolic space action, challenging assumptions about element behavior in such groups.

Contribution

It introduces a new example of a Morse element that is not loxodromic or WPD, expanding understanding of Morse elements in geometric group theory.

Findings

Constructed a Morse local-to-global group with a non-loxodromic Morse element

Demonstrated the existence of Morse elements that are not WPD

Used small-cancellation techniques in the construction

Abstract

We use small-cancellation techniques to construct a Morse local-to-global group G with an infinite-order Morse element that is not loxodromic in any action of G on a hyperbolic space. In particular, the element cannot be WPD.