Lattices, injective metrics and the K(π,1) conjecture

Haettel, Thomas

doi:10.2140/agt.2024.24.4007

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Lattices, injective metrics and the $K(\pi,1)$ conjecture

Thomas Haettel

Algebraic & Geometric Topology 24 (2024) 4007–4060

DOI: 10.2140/agt.2024.24.4007

Abstract

Starting with a lattice with an action of $ℤ$ or $ℝ$ , we build a Helly graph or an injective metric space. We deduce that the $ℓ^{\infty}$ 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 $ℓ^{\infty}$ metric on any Euclidean building of type $Ã_{n}$ extended, ${\tilde{B}}_{n}$ , ${\tilde{C}}_{n}$ or ${\tilde{D}}_{n}$ is injective, and its thickening is a Helly graph.

Concerning Artin groups of Euclidean types $Ã_{n}$ and ${\tilde{C}}_{n}$ , we show that the natural piecewise $ℓ^{\infty}$ 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.

Keywords

injective metrics, Helly graphs, lattices, Artin groups, Euclidean buildings, Cartan–Hadamard

Mathematical Subject Classification

Primary: 20E42, 05B35, 52A35, 06A12

References

Publication

Received: 15 February 2023

Revised: 12 September 2023

Accepted: 23 October 2023

Published: 9 December 2024

Authors

Thomas Haettel
Institut Montpelliérain Alexander Grothendieck Université de Montpellier, CNRS Montpellier France	IRL 3457, CRM-CNRS Université de Montréal Montréal Canada

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Volume 24, issue 7 (2024)