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Asphericity of cubical presentations: the general case

Macarena Arenas

Abstract

We show that, under suitable hypotheses, the coned-off spaces associated to $C(9)$ cubical small-cancellation presentations are aspherical, and use this to provide classifying spaces, or classifying spaces for proper actions, for their fundamental groups. Along the way, we show that the Cohen--Lyndon property holds for the subgroups of the fundamental group of a non-positively curved cube complex associated to a $C(9)$ cubical presentation, and thus obtain near-sharp upper and lower bounds for the (rational) cohomological dimension of these quotients. We apply these results to give an alternative construction of compact $K(π,1)$ for Artin groups with no labels in $\{3,4\}$, from which a new direct sum decomposition for their homology and cohomology with various coefficients above a certain dimension follows. We also address a question of Wise about the virtual torsion-freeness of cubical small-cancellation groups.

Asphericity of cubical presentations: the general case

Abstract

We show that, under suitable hypotheses, the coned-off spaces associated to cubical small-cancellation presentations are aspherical, and use this to provide classifying spaces, or classifying spaces for proper actions, for their fundamental groups. Along the way, we show that the Cohen--Lyndon property holds for the subgroups of the fundamental group of a non-positively curved cube complex associated to a cubical presentation, and thus obtain near-sharp upper and lower bounds for the (rational) cohomological dimension of these quotients. We apply these results to give an alternative construction of compact for Artin groups with no labels in , from which a new direct sum decomposition for their homology and cohomology with various coefficients above a certain dimension follows. We also address a question of Wise about the virtual torsion-freeness of cubical small-cancellation groups.

Paper Structure

This paper contains 24 sections, 45 theorems, 43 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Applications
  3. Theorem \ref{['thm:clintro']} and the Cohen-Lyndon property for triples
  4. Strategy and structure
  5. Background
  6. Cubical small-cancellation notions
  7. Diagrams in 2-complexes and cubical presentations
  8. Cohomology and the Cohen-Lyndon property
  9. Disassembling the cubical part of $\widetilde{X^*}$
  10. Intersections, intersections, intersections
  11. Reassembling the cubical part of $\widetilde{X^*}$ (carefully)
  12. The Cohen-Lyndon property for cubical C(9) presentations
  13. Asphericity
  14. Asphericity of the reduced space
  15. Asphericity for proper actions
  16. ...and 9 more sections

Key Result

Theorem 1.1

[Theorem thm:asph] Let $X^*=\langle X \mid \{Y_1, \ldots, Y_n\} \rangle$ be a cubical presentation. If $X^*$ satisfies the cubical $C(9)$ condition, then $\bar{X}^*$ is aspherical. In particular, if $X^*$ is reduced, then $X^*$ is a classifying space for $\pi_1 X^*$.

Figures (5)

  • Figure 1: Impossible diagrammatic features in the proof of connectedness in Proposition \ref{['prop: conecell ints']}.
  • Figure 2: Heuristics of the relevant disc diagrams in the proofs of the various items in Lemma \ref{['lem:special like behaviours']}. From left to right: hyperplane carriers are simply-connected, embedded, and their pairwise non-empty intersections are simply-connected. Impossible cornsquares and shells are indicated in orange; possible cornsquares are marked in green.
  • Figure 3: The objects considered in the proof of Proposition \ref{['prop:helly']}.
  • Figure 4: A thick annuladder.
  • Figure 5: The cubical presentation $X^*$ is obtained by gluing squares and 3-cubes to the 1-skeleton of a truncated cuboctahedron. $X^*$ satisfies the $C(8)$ condition but not Greendlinger's Lemma, as can be seen by considering the minimal complexity disc diagram $D$ in the right of the figure.

Theorems & Definitions (98)

  • Theorem 1.1
  • Corollary 1.2
  • Theorem 1.3
  • Lemma 1.4
  • Theorem 1.5
  • Corollary 1.6
  • Theorem 1.7
  • Definition 2.2: Elevations
  • Definition 2.3: Pieces
  • Definition 2.4
  • ...and 88 more