The Deligne Complex for the $B_3$ Artin Group

TL;DR

The paper proves that the piecewise Euclidean Moussong metric on the Deligne complex of the 3-dimensional Artin group of type $B_3$ is CAT$(0)$ by establishing a combinatorial CAT$(1)$ criterion for $B_3$-complexes and verifying it through a detailed analysis of short loops. The authors develop a Bowditch-style shrinkability framework, reduce loops to edge paths, and perform a cycle-by-cycle verification (4-, 6-, 8-, and 10-cycles) in $D(B_3)$, leveraging development to $C(B_3)$ and a mapping-class-group-based embedding of $A(B_n)$ into $A(A_{2n-1})$. This leads to CAT$(1)$ for the spherical Deligne complex of type $B_3$, and consequently CAT$(0)$ for the Moussong metric on the Deligne complex when the ambient group has no subdiagram of type $H_3$. The results advance the program of establishing nonpositive curvature and the $K(\pi,1)$ property for a broad class of Artin groups, and provide a transferable framework for analyzing other 2- and 3-dimensional cases. The techniques combine combinatorial curvature criteria, edge-path reductions, and geometric embeddings via mapping class groups to push toward a general CAT$(0)$ conjecture for 3-dimensional Artin groups.

Abstract

We show that the piecewise Euclidean Moussong metric on the Deligne complex of the Artin group of type $B_3$ is $\mathrm{CAT}(0)$. We do this by establishing a criteria for a complex made of $B_3$ simplices to be $\mathrm{CAT}(1)$ in terms of embedded edge paths, which in particular applies to the spherical Deligne complex of type $B_3$. This provides one more step to showing that the Moussong metric is $\mathrm{CAT}(0)$ for any 3-dimensional Artin group.

Figures (19)

• Figure 1: New Coxeter-Dynkin diagrams
• Figure 2: Some short closed edge paths
• Figure 3: Filled edge paths
• Figure 4: A non-admissible loop
• Figure 5: Developing a geodesic through a vertex of type $\hat{s}_2$

Theorems & Definitions (63)

• Theorem 1.1
• Theorem 1.1
• Example 1.1
• Theorem 1.1
• Theorem 2.1
• Proposition 2.2
• Lemma 2.3
• proof
• Lemma 2.4
• Lemma 2.5