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

Carolyn Abbott, Stefanie Zbinden

TL;DR

The paper constructs a $C'(1/9)$-group that is Morse local-to-global yet contains an infinite-order Morse element not loxodromic in any action on a hyperbolic space, and moreover not a loxodromic WPD element. By starting from a $C'(4/N)$-group and adding generators and relators that embed large powers of a new Morse element $a$, the authors enforce non-hyperbolic behavior in all hyperbolic-space actions while preserving MLTG. A key geometric tool is an embedded-cycle lemma in hyperbolic spaces, ensuring subsegments of cycles have endpoints close together, which is used to force non-loxodromicity of $a$ via relator geometry. The result demonstrates that Morse elements in MLTG groups need not yield acylindrical hyperbolicity and highlights subtle distinctions between MLTG, IPSC, and boundary behavior in small-cancellation groups.

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.

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

TL;DR

The paper constructs a -group that is Morse local-to-global yet contains an infinite-order Morse element not loxodromic in any action on a hyperbolic space, and moreover not a loxodromic WPD element. By starting from a -group and adding generators and relators that embed large powers of a new Morse element , the authors enforce non-hyperbolic behavior in all hyperbolic-space actions while preserving MLTG. A key geometric tool is an embedded-cycle lemma in hyperbolic spaces, ensuring subsegments of cycles have endpoints close together, which is used to force non-loxodromicity of via relator geometry. The result demonstrates that Morse elements in MLTG groups need not yield acylindrical hyperbolicity and highlights subtle distinctions between MLTG, IPSC, and boundary behavior in small-cancellation groups.

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.

Paper Structure

This paper contains 10 sections, 13 theorems, 20 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The Morse local-to-global property
  4. Small cancellation
  5. Morse geodesics in small-cancellation groups
  6. Increasing partial small-cancellation condition
  7. Embedded cycles in hyperbolic spaces
  8. Construction of the groups
  9. Small cancellation and Morse local-to-global properties
  10. Proof of Main Theorem

Key Result

Theorem 1.1

There exists a Morse local-to-global group $G$ and an infinite-order Morse element $a\in G$ such that $a$ is not loxodromic in any action of $G$ on a hyperbolic space. In particular, the element $a$ is not a loxodromic WPD element in any such action.

Figures (2)

  • Figure 1: Possible configurations of $a_1,a_1'$, and $u_1$ in the proof of Lemma \ref{['lem:shortsubpaths']}. Finding $a_2,a_2',u_2$ is similar.
  • Figure 2: Finding the desired subpath $\lambda'$ (blue) in the proof of Lemma \ref{['lem:shortsubpaths']} when $d(y,u_1)<d(y,u_2)+2D$.

Theorems & Definitions (33)

  • Theorem 1.1
  • Definition 2.1: Morse local-to-global
  • Definition 2.2: Piece
  • Definition 2.3: $C'(\lambda)$ condition
  • Definition 2.4: Intersection function
  • Lemma 2.5: arzhantseva2019negative
  • Lemma 2.6: Z:small_cancellation
  • Definition 2.7
  • Definition 2.8
  • Remark 2.9
  • ...and 23 more