[Paper Review] On regular CAT(0) cube complexes

Nir Lazarovich|arXiv (Cornell University)|Nov 1, 2014

Geometric and Algebraic Topology10 references3 citations

TL;DR

This paper establishes a necessary and sufficient combinatorial condition—superstar-transitivity—for the uniqueness of CAT(0) cube complexes with a given finite flag simplicial complex $L$ as vertex links. It proves that such complexes have the hyperplane automorphism extension property and, under additional conditions, their automorphism groups are virtually simple, generalizing classical results on trees and buildings.

ABSTRACT

We provide a necessary and sufficient condition on a finite flag simplicial complex, L, for which there exists a unique CAT(0) cube complex whose vertex links are all isomorphic to L. We then find new examples of such CAT(0) cube complexes and prove that their automorphism groups are virtually simple. The latter uses a result, which we prove in the appendix, about the simplicity of certain subgroups of the automorphism group of a rank-one CAT(0) cube complex. This result generalizes previous results by Tits and by Haglund and Paulin.

Motivation & Objective

• To determine when a CAT(0) cube complex with a fixed finite flag simplicial complex $L$ as vertex links is uniquely determined up to isomorphism.
• To characterize the automorphism groups of such unique complexes, particularly their simplicity properties.
• To generalize Tits' and Haglund-Paulin's results on simplicity of automorphism groups in rank-one CAT(0) cube complexes.
• To establish the hyperplane automorphism extension property (HAEP) in unique $L$-cube-complexes, enabling group-theoretic analysis.

Proposed method

• Introduce the notion of superstar-transitivity: for any two simplices $\sigma, \sigma'$ in $L$ and any link-isomorphism $\phi: \text{st}_L(\sigma) \to \text{st}_L(\sigma')$ sending $\sigma$ to $\sigma'$, there exists a global automorphism $\Phi$ of $L$ extending $\phi$.
• Use an inductive construction on vertices of the cube complex to build isomorphisms between $L$-cube-complexes, relying on the superstar-transitivity condition.
• Construct the Davis complex $D(L)$ as a canonical example of an $L$-cube-complex, and prove it is the unique such complex if and only if $L$ is superstar-transitive.
• Prove the hyperplane automorphism extension property (HAEP): any automorphism of a finite collection of pairwise transverse hyperplanes extends to the whole complex.
• Apply the rank rigidity theorem and a normal subgroup analysis (Proposition A.8) to show that non-trivial normal subgroups of the automorphism group act non-trivially on the boundary and are non-elementary.
• Use the HAEP and the simplicity result from the appendix (generalizing Tits and Haglund-Paulin) to conclude that the automorphism group is virtually simple under suitable conditions.

Experimental results

Research questions

• RQ1 Under what combinatorial conditions on a finite flag simplicial complex $L$ does there exist a unique CAT(0) cube complex with vertex links isomorphic to $L$?
• RQ2 Does the hyperplane automorphism extension property (HAEP) hold for all unique $L$-cube-complexes, and what does it imply for the automorphism group?
• RQ3 When is the automorphism group of a unique $L$-cube-complex virtually simple, and how does this generalize known results on trees and buildings?
• RQ4 What role does superstar-transitivity play in ensuring both uniqueness and group-theoretic rigidity in $L$-cube-complexes?
• RQ5 How do normal subgroups of the automorphism group of a rank-one CAT(0) cube complex behave, and what conditions force them to be non-elementary and non-trivial?

Key findings

• A finite flag simplicial complex $L$ admits a unique CAT(0) cube complex with vertex links isomorphic to $L$ if and only if $L$ is superstar-transitive.
• The Kneser complex $K_n^d$ provides a new family of examples of unique CAT(0) $L$-cube-complexes of arbitrary dimension.
• In any unique $L$-cube-complex, every automorphism of a finite collection of pairwise transverse hyperplanes extends to an automorphism of the entire complex (HAEP).
• The automorphism group of a unique $L$-cube-complex is virtually simple if the complex is rank-one and the link $L$ satisfies the conditions in the appendix.
• The appendix proves a general simplicity result: for a proper CAT(0) space $X$, if $G^+$ is a non-elementary group containing a rank-one isometry and $N \triangleleft G^+$ acts non-trivially on $\partial X$, then $N$ is non-elementary and $\Lambda(N) = \partial X$, leading to virtual simplicity of $\text{Aut}(X)$ under suitable conditions.
• The result generalizes Tits' and Haglund-Paulin's theorems on simplicity of halfspace fixator groups in automorphism groups of CAT(0) cube complexes.