[Paper Review] Lattices, injective metrics and the $K(π,1)$ conjecture

Thomas Haettel|arXiv (Cornell University)|Sep 16, 2021

Geometric and Algebraic Topology4 citations

TL;DR

This paper establishes that the $ι^\infty$-metric on Euclidean buildings and Deligne complexes of types $\widetilde{A_n}$, $\widetilde{C_n}$, $\widetilde{B_n}$, and $\widetilde{D_n}$ is injective, and their thickenings are Helly graphs. Using a general construction from lattices with $Ω$-actions, it proves that Garside groups and FC-type Artin groups act on injective metric spaces, yielding a metric proof of the $K(\pi,1)$ conjecture and new results on bicombings, parabolic subgroups, and fixed-point subgroups.

ABSTRACT

Starting with a lattice with an action of $\mathbb{Z}$ or $\mathbb{R}$, we build a Helly graph or an injective metric space. We deduce that the $\ell^\infty$ orthoscheme complex of any bounded graded lattice is injective. We also prove a Cartan-Hadamard result for locally injective metric spaces. We apply this to show that any Garside group or any FC type Artin group acts on an injective metric space and on a Helly graph. We also deduce that the natural piecewise $\ell^\infty$ metric on any Euclidean building of type $ ilde{A_n}$ extended, $ ilde{B_n}$, $ ilde{C_n}$ or $ ilde{D_n}$ is injective, and its thickening is a Helly graph. Concerning Artin groups of Euclidean types $ ilde{A_n}$ and $ ilde{C_n}$, we show that the natural piecewise $\ell^\infty$ metric on the Deligne complex is injective, the thickening is a Helly graph, and it admits a convex bicombing. This gives a metric proof of the $K(π,1)$ conjecture, as well as several other consequences usually known when the Deligne complex has a CAT(0) metric.

Motivation & Objective

• To develop a general method for constructing injective metric spaces and Helly graphs from lattices with $Ω$-actions.
• To prove that the $ι^\infty$-metric on Euclidean buildings of types $\widetilde{A_n}$, $\widetilde{B_n}$, $\widetilde{C_n}$, and $\widetilde{D_n}$ is injective.
• To show that the Deligne complex of Euclidean Artin groups of types $\widetilde{A_n}$ and $\widetilde{C_n}$ admits a convex, consistent, and $A$-equivariant bicombing.
• To establish that the $K(\pi,1)$ conjecture holds for these Artin groups via a metric approach.

Proposed method

• Constructing a metric $d(x,y) = \inf\{t \in H_+ \mid f_{-t}(x) \leq y \leq f_t(x)\}$ on a lattice $L$ with an order-preserving action of $H = \mathbb{Z}$ or $\mathbb{R}$, where $f_t$ is a continuous increasing $H$-action.
• Proving that this metric yields a Helly graph when $H = \mathbb{Z}$ and an injective metric space when $H = \mathbb{R}$, under the condition that upperly bounded sets have joins.
• Applying the construction to the lattice of flats in Euclidean buildings and to the poset of parabolic subgroups in Artin groups.
• Showing that the natural piecewise-$\ell^\infty$ metric on the Deligne complex of $\widetilde{A_n}$ and $\widetilde{C_n}$ is injective and admits a convex, consistent, and $A$-equivariant geodesic bicombing.
• Using the existence of such a bicombing to deduce contractibility of the Deligne complex and thus verify the $K(\pi,1)$ conjecture.
• Extending results on parabolic subgroups, centralizers, and fixed-point subgroups by leveraging the $A$-equivariant bicombing and $σ$-stability of subcomplexes.

Experimental results

Research questions

• RQ1 Can the $K(\pi,1)$ conjecture for Euclidean Artin groups be proven using injective metric geometry rather than CAT(0) geometry?
• RQ2 Is the natural piecewise-$\ell^\infty$ metric on Euclidean buildings of types $\widetilde{A_n}$, $\widetilde{B_n}$, $\widetilde{C_n}$, and $\widetilde{D_n}$ injective?
• RQ3 Does the Deligne complex of Euclidean Artin groups of types $\widetilde{A_n}$ and $\widetilde{C_n}$ admit a convex, consistent, and $A$-equivariant geodesic bicombing?
• RQ4 Can results on parabolic subgroups, centralizers, and fixed-point subgroups of Artin groups be extended using the existence of a convex bicombing on the Deligne complex?
• RQ5 What is the relationship between the Helly property, injective metrics, and group actions in the context of Garside and Artin groups?

Key findings

• The natural piecewise-$\ell^\infty$ metric on any Euclidean building of type $\widetilde{A_n}$, $\widetilde{B_n}$, $\widetilde{C_n}$, or $\widetilde{D_n}$ is injective.
• The thickening of the vertex set of such buildings is a Helly graph.
• The Deligne complex of Euclidean Artin groups of type $\widetilde{A_n}$ or $\widetilde{C_n}$ admits a convex, consistent, and $A$-equivariant geodesic bicombing.
• The Deligne complex is contractible, providing a metric proof of the $K(\pi,1)$ conjecture for these groups.
• The intersection of any family of parabolic subgroups in these Artin groups is itself a parabolic subgroup.
• Fixed-point subgroups of symmetric actions on the Artin system are isomorphic to Artin groups of lower rank, generalizing results of Digne and Crisp.