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Hyperbolicity in non-metric cubical small-cancellation

Macarena Arenas, Kasia Jankiewicz, Daniel T. Wise

Abstract

Given a non-positively curved cube complex $X$, we prove that the quotient of $π_1X$ defined by a cubical presentation $\langle X\mid Y_1,\dots, Y_s\rangle$ satisfying sufficient non-metric cubical small-cancellation conditions is hyperbolic provided that $π_1X$ is hyperbolic. This generalises the fact that finitely presented classical $C(7)$ small-cancellation groups are hyperbolic.

Hyperbolicity in non-metric cubical small-cancellation

Abstract

Given a non-positively curved cube complex , we prove that the quotient of defined by a cubical presentation satisfying sufficient non-metric cubical small-cancellation conditions is hyperbolic provided that is hyperbolic. This generalises the fact that finitely presented classical small-cancellation groups are hyperbolic.
Paper Structure (13 sections, 11 theorems, 6 equations, 7 figures)

This paper contains 13 sections, 11 theorems, 6 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Proof strategy
  3. Structure of the paper
  4. Cubical background
  5. Non-positively curved cube complexes
  6. Cubical presentations
  7. Disc diagrams in $X^*$
  8. Cubical small-cancellation
  9. Greendlinger's Lemma
  10. Hyperbolic background
  11. Thin bigon criterion for hyperbolicity
  12. The piece metric
  13. Proof of hyperbolicity

Key Result

Lemma 2.3

Let $D\to X^*$ be a disc diagram, then there exist disc diagrams $D'\to X^*$ and $D"\to X^*$ satisfying:

Figures (7)

  • Figure 1: Cone-cells in a disc diagram. In figures we will often omit the cell structure of cone-cells, unless needed.
  • Figure 2: The six reduction moves from Definition \ref{['defn:reduction moves']}.
  • Figure 3: Blue paths are contiguous pieces, and yellow paths are pieces but not contiguous pieces.
  • Figure 4: Example of a ladder.
  • Figure 5: Steps of the proof of Lemma \ref{['lem: piece length bound']}.
  • ...and 2 more figures

Theorems & Definitions (24)

  • proof
  • Definition 2.1: Reduction moves
  • Definition 2.2: Reduced and weakly reduced disc diagram
  • Lemma 2.3
  • Remark 2.4
  • Lemma 2.5
  • Definition 2.6: Ladder
  • Lemma 2.7: Cubical Greendlinger's Lemma WiseIsraelHierarchyJankiewicz17
  • Proposition 3.1: Thin Bigon Criterion
  • Definition 4.1
  • ...and 14 more