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Relative Cubulation of Small Cancellation Free Products

Eduard Einstein, Thomas Ng

Abstract

We expand the class of groups with relatively geometric actions on CAT(0) cube complexes by proving that it is closed under $C'(\frac16)$--small cancellation free products. We build upon a result of Martin and Steenbock who prove an analogous result in the more specialized setting of groups acting properly and cocompactly on CAT(0) cube complexes. Our methods make use of the same blown-up complex of groups to construct a candidate collection of walls. However, rather than arguing geometrically, we show relative cubulation by appealing to a boundary separation criterion and proving that wall stabilizers form a sufficiently rich family of full relatively quasiconvex codimension-one subgroups. The additional flexibility of relatively geometric actions has surprising new applications. In particular, we prove that $C'(\frac16)$--small cancellation free products of residually finite groups are residually finite.

Relative Cubulation of Small Cancellation Free Products

Abstract

We expand the class of groups with relatively geometric actions on CAT(0) cube complexes by proving that it is closed under --small cancellation free products. We build upon a result of Martin and Steenbock who prove an analogous result in the more specialized setting of groups acting properly and cocompactly on CAT(0) cube complexes. Our methods make use of the same blown-up complex of groups to construct a candidate collection of walls. However, rather than arguing geometrically, we show relative cubulation by appealing to a boundary separation criterion and proving that wall stabilizers form a sufficiently rich family of full relatively quasiconvex codimension-one subgroups. The additional flexibility of relatively geometric actions has surprising new applications. In particular, we prove that --small cancellation free products of residually finite groups are residually finite.

Paper Structure

This paper contains 32 sections, 56 theorems, 15 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Structure of this paper:
  3. Background
  4. Fine hyperbolic graphs and relative hyperbolicity
  5. Relatively geometric actions on $\operatorname{CAT}(0)$ cube complexes
  6. Small cancellation free products
  7. A complex of groups for $G$ and the construction of its development
  8. The developed complex $X$, via complex of groups structure for $G$
  9. The relatively hyperbolic structure for $G$ with respect to the vertex groups
  10. Acylindricity of relative hyperbolic group actions
  11. Small cancellation blow-up
  12. The Construction of the Walls
  13. $\Omega$--hyperstructures
  14. Hypergraphs coming from the subdivided complex $X_{bal}$
  15. The hyperstructures of $\mathcal{E}G_{bal}$
  16. ...and 17 more sections

Key Result

Theorem 1.1

Let $A_1, \dotsc, A_n$ be finitely generated residually finite groups with $n \geq 2$. Any $C'(\frac{1}{6})$--small cancellation free product of $A_1, \dotsc, A_n$ by a finite set of relators is residually finite.

Figures (6)

  • Figure 1: The building blocks of $G(\mathcal{K})$. The underlying complex of $G(\mathcal{K})$ is built from the disjoint union of the two edge graph $L'$ on the right and a polygon consisting of $2N$ copies of the triangle on the left glued together so that the apex is at the center of the polygon. The polygon and $L'$ are glued together as shown.
  • Figure 2: A blown-up polygon in $\mathcal{E}G$ and its image in the polygonal complex $X$
  • Figure 3: The blow-up described in \ref{['simple example']} is the standard tree of spaces over the Bass-Serre tree
  • Figure 4: On the right is a square complex with a hyperplane drawn in red. On the left is its abstract carrier.
  • Figure 5: Commutative diagram depicting the various subgroups considered in the proof of \ref{['hyp stab full']}.
  • ...and 1 more figures

Theorems & Definitions (123)

  • Theorem 1.1
  • Theorem 1.2
  • Corollary 1.3
  • Definition 2.1: BowditchRH, written as stated in Hruska2010
  • Theorem 2.2: WiseMP10
  • Definition 2.4
  • Definition 2.5
  • Definition 2.6
  • Theorem 2.7
  • Theorem 2.8: RelGeom, see also GM20
  • ...and 113 more