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Cubulating random quotients of hyperbolic cubulated groups

David Futer, Daniel T. Wise

Abstract

We show that low-density random quotients of cubulated hyperbolic groups are again cubulated (and hyperbolic). Ingredients of the proof include cubical small-cancellation theory, the exponential growth of conjugacy classes, and the statement that hyperplane stabilizers grow exponentially more slowly than the ambient cubical group.

Cubulating random quotients of hyperbolic cubulated groups

Abstract

We show that low-density random quotients of cubulated hyperbolic groups are again cubulated (and hyperbolic). Ingredients of the proof include cubical small-cancellation theory, the exponential growth of conjugacy classes, and the statement that hyperplane stabilizers grow exponentially more slowly than the ambient cubical group.

Paper Structure

This paper contains 22 sections, 30 theorems, 86 equations, 13 figures.

Table of Contents

  1. Introduction
  2. Main result
  3. Density and optimality of constants
  4. Actions on other metric spaces
  5. Overview
  6. Acklowledgements
  7. Growth
  8. Cubical presentations and small-cancellation theory
  9. Cubical presentations and pieces
  10. Wallspace small-cancellation conditions
  11. The properness criterion
  12. Controlling pieces in generic quotients
  13. Loose pieces and convex hulls of quasi-axes
  14. Wall-pieces
  15. Cone-pieces between two closed--geodesics
  16. ...and 7 more sections

Key Result

Theorem 1.1

Let $G=\pi_1X$, where $X$ is a compact non-positively curved cube complex, and suppose that $G$ is hyperbolic. Let $b$ be the growth exponent of $G$ with respect to the universal cover $\widetilde{X}$. Let $a$ be the maximal growth exponent of a stabilizer of an essential hyperplane of $\widetilde{X Then with overwhelming probability as $\ell \to \infty$, for any set of conjugacy classes $[g_1], \

Figures (13)

  • Figure 1: A cubical presentation illustrating conditions \ref{['B6:wallspace']} and \ref{['B6:aut']} of \ref{['Def:B6']}. In this example, as in \ref{["Thm:C'20Proper"]}, we have $\pi_1 Y_i \cong \mathbb{Z}$ for each $Y_i$, and the wallspace structure on $Y_i$ has two diametrically opposed hyperplanes in each wall. Unlike the setting of \ref{["Thm:C'20Proper"]}, $\pi_1 X$ is not hyperbolic in this example.
  • Figure 2: The scenario that arises in hypothesis \ref{['Itm:Separation']} of \ref{['Thm:cubulating X star']}. For the geodesic $\kappa$, we have to verify that neither $p$ nor $q$ is $2$--proximate to $U_2 = u_2 \cup u_2'$.
  • Figure 3: The terminology and setup of \ref{['Def:OverlapOrientation']}. The $J$--loose piece $s = [p,q]$ is shown in yellow, and the companion $s' = [p',q']$ in orange. In this example, $h$ reverses orientation on the piece $s$, and $s$ has nontrivial $h$--overlap $[p, h^{-1}p']$.
  • Figure 4: Bottom: $y_1 y_2$ is a geodesic, and $h y_1$ fellow-travels $y_2$. Hence there is a bounded amount of fellow-travelling between $y_1$ and $h y_1$. This implies $(y_1 x_1)^\infty$ is a quasigeodesic, and similarly for $(y_3 x_3)^\infty$. The two quasigeodesics must $\delta'$--fellow-travel, hence there is a geodesic of length $\delta'$ from the endpoint of $y_3$ to some point on $(y_1 x_1)^\infty$.
  • Figure 5: The figure lives in $\Upsilon$. The blue quasi-geodesics $\widetilde{\varpi}_i$ and $\widetilde{\varpi}_j$ are the images of axes in $\widetilde{X}$. The black geodesics are axes $\widetilde{\gamma}_i$ and $\widetilde{\gamma}_j$, respectively. Since $\widetilde{\varpi}_i$ must $\eta$--fellow-travel with $\gamma_i$, there is a point $x_i"$ that is $\eta$--close to $x_i'$, and similarly for the others. The $2\delta$--loose cone-piece between $\widetilde{\gamma}_i$ and $\widetilde{\gamma}_j$ will contain the segment $[x_i"', y_i"']$.
  • ...and 8 more figures

Theorems & Definitions (67)

  • Theorem 1.1
  • Corollary 1.2
  • Theorem 1.3
  • Corollary 1.4
  • Definition 2.1: Translation lengths
  • Definition 2.2: Growth
  • Theorem 2.3
  • Theorem 2.4
  • proof
  • Remark 2.5
  • ...and 57 more