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A cubulation with no factor system

Sam Shepherd

TL;DR

The paper demonstrates that the standard route to proving hierarchical hyperbolicity for cubulated groups via a factor system can fail: it constructs a cubulated group with no factor system and a separate example where the induced action on the contact graph is not acylindrical. The first construction uses an anti-torus in a product of trees and attaches infinite strips to create a CAT(0) cube complex $X$ on which the group acts; this yields obstructions to any factor system and violates three known sufficient conditions for having one. The second construction passes to an HNN extension, building a tree-of-spaces cube complex $Y$ in which the action on the contact graph $\mathcal{C}Y$ is not acylindrical, again implying no factor system. Together, these results highlight limitations in extending HHG structure to all cubulated groups and raise questions about the existence of largest acylindrical actions and the status of the second group as an HHG. The paper also shows that the largest acylindrical action for the second group occurs on the tree of cylinders $T_c$, separating these phenomena from potential hierarchical hyperbolicity.

Abstract

The primary method for showing that a given cubulated group is hierarchically hyperbolic is by constructing a factor system on the cube complex. In this paper we show that such a construction is not always possible, namely we construct a cubulated group for which the cube complex does not have a factor system. We also construct a cubulated group for which the induced action on the contact graph is not acylindrical.

A cubulation with no factor system

TL;DR

The paper demonstrates that the standard route to proving hierarchical hyperbolicity for cubulated groups via a factor system can fail: it constructs a cubulated group with no factor system and a separate example where the induced action on the contact graph is not acylindrical. The first construction uses an anti-torus in a product of trees and attaches infinite strips to create a CAT(0) cube complex on which the group acts; this yields obstructions to any factor system and violates three known sufficient conditions for having one. The second construction passes to an HNN extension, building a tree-of-spaces cube complex in which the action on the contact graph is not acylindrical, again implying no factor system. Together, these results highlight limitations in extending HHG structure to all cubulated groups and raise questions about the existence of largest acylindrical actions and the status of the second group as an HHG. The paper also shows that the largest acylindrical action for the second group occurs on the tree of cylinders , separating these phenomena from potential hierarchical hyperbolicity.

Abstract

The primary method for showing that a given cubulated group is hierarchically hyperbolic is by constructing a factor system on the cube complex. In this paper we show that such a construction is not always possible, namely we construct a cubulated group for which the cube complex does not have a factor system. We also construct a cubulated group for which the induced action on the contact graph is not acylindrical.
Paper Structure (3 sections, 7 theorems, 4 equations, 5 figures)

This paper contains 3 sections, 7 theorems, 4 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Example with no factor system
  3. Example with non-acylindrical action on the contact graph

Key Result

Theorem 1.2

There is a cubulated group $G\curvearrowright X$ such that $X$ does not have a factor system.

Figures (5)

  • Figure 1: Rectangle in $\Pi$ with two sides labeled by $w_1^n$ and $w_2^i$ that meet at the vertex $(p_1,p_2)$.
  • Figure 2: Rectangle in $\Pi$ with two sides labeled by $w_1^k$ and $w_2^j$ that meet at the vertex $(p_1,p_2)$.
  • Figure 3: The $w_1$-geodesics $F$ and $F'$ with their attached strips.
  • Figure 4: The positioning of $g_n(p_1,p_2)$.
  • Figure 5: The arrangement of the $w_1$-geodesics $g_n^i F$ and $g_n^i F'$.

Theorems & Definitions (17)

  • Definition 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Lemma 2.1
  • proof
  • Theorem 2.2
  • proof
  • Remark 2.3
  • Remark 2.4
  • Theorem 3.1
  • ...and 7 more