Homotopy equivalent boundaries of cube complexes

Abstract

A finite-dimensional CAT(0) cube complex $X$ is equipped with several well-studied boundaries. These include the Tits boundary (which depends on the CAT(0) metric), the Roller boundary (which depends only on the combinatorial structure), and the simplicial boundary (which also depends only on the combinatorial structure). We use a partial order on a certain quotient of the Roller boundary to obtain the simplicial Roller boundary. Then, we show that the Tits, simplicial, and simplicial Roller boundaries are all homotopy equivalent, $Aut(X)$--equivariantly up to homotopy. As an application, we deduce that the perturbations of the CAT(0) metric introduced by Qing do not affect the equivariant homotopy type of the Tits boundary. Along the way, we develop a self-contained exposition providing a dictionary among different perspectives on cube complexes.

Figures (7)

• Figure 1: Left: a linear staircase. Right: a sublinear staircase. For both staircases, the simplicial boundary is a $1$--simplex, whose vertices correspond to the UBS $\mathcal{V}$ of vertical hyperplanes and the UBS $\mathcal{H}$ of horizontal hyperplanes. In both staircases, $\mathcal{H}$ is not visible (in either the $\ell^1$ or the $\ell^2$ metric), because any geodesic ray crossing $\mathcal{H}$ must also cross $\mathcal{V}$. In the sublinear staircase, $\mathcal{V} \sqcup \mathcal{H}$ is also not $\ell^2$--visible.
• Figure 2: Part of a CAT(0) cube complex for which the set of hyperplanes is a UBS. Each horizontal hyperplane is the base of a chain of horizontal hyperplanes whose inseparable closure consists of all but finitely many of the horizontal hyperplanes. Each horizontal hyperplane crosses all but finitely many vertical ones. So each horizontal hyperplane is dominant. The vertical hyperplanes are not dominant. Indeed, if $v$ is vertical, it is the base of a chain of vertical hyperplanes whose inseparable closure consists of vertical hyperplanes, while $v$ fails to cross infinitely many horizontal hyperplanes. On the other hand, $v$ is also the base of a sequence of hyperplanes consisting of $v$ and infinitely many horizontal hyperplanes, whose canonical half-spaces form a descending chain. The inseparable closure of this set of hyperplanes contains all but finitely many of the hyperplanes, and therefore is not a minimal UBS, so this set is not a chain of hyperplanes.
• Figure 3: The sets $\mathcal{U}'_i,\mathcal{U}_j$ in the proof of Lemma \ref{['Lem:DominantChain']}.\ref{['item:dominants_cross']}.
• Figure 4: The cube complex $X$ in Example \ref{['exmp:weird_simplex']}, whose simplicial boundary has a "weird" $1$--simplex. The hyperplanes $B_i$ are shown in broken lines, the hyperplanes $S_j$ are solid, and the hyperplanes $D_k$ are dotted. This non-perspective drawing represents an embedding into $\mathbb{R}^3$ where the solid hyperplanes have also been rescaled by a factor of 2.
• Figure 5: The figure shows a portion of the cube $C_{y_n}$, with the back wall is $M_n = C_{y_n} \cap \hat{h}_n$. The angle between $\alpha_{y_n}'$ and $M_n$ is at least $\kappa$, so the angle between $\eta_n$ and $M_n$ is bounded away from $0$ by $\kappa_1 = \kappa (1-s)$. But the angle between $\gamma_{y_n}'$ and $\eta_n$ can be made much smaller than $\kappa (1-s)$ by taking $n$ large. So $\gamma_{y_n}'$ makes a positive angle with $\hat{h}_n$.

Theorems & Definitions (215)

• Definition 1.1
• Theorem A
• Theorem B
• Remark 1.2
• Definition 2.1: Weak topology
• Definition 2.2: Metric topology
• Definition 2.4: Simplicial realization
• Definition 2.5: Nerve
• Lemma 2.6: Recognizing homotopic maps
• proof