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 $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>ℤ</mi></math>$ or $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>ℝ</mi></math>$ , we build a Helly graph or an injective metric space. We deduce that the $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msup><mrow><mi>ℓ</mi></mrow><mrow><mi>∞</mi></mrow></msup></math>$ 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 $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msup><mrow><mi>ℓ</mi></mrow><mrow><mi>∞</mi></mrow></msup></math>$ metric on any Euclidean building of type $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mrow><mi>Ã</mi></mrow><mrow><mi>n</mi></mrow></msub></math>$ extended, $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mrow><mover accent="true"><mrow><mi>B</mi></mrow><mo accent="true">~</mo></mover></mrow><mrow><mi>n</mi></mrow></msub></math>$ , $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mrow><mover accent="true"><mrow><mi>C</mi></mrow><mo accent="true">~</mo></mover></mrow><mrow><mi>n</mi></mrow></msub></math>$ or $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mrow><mover accent="true"><mrow><mi>D</mi></mrow><mo accent="true">~</mo></mover></mrow><mrow><mi>n</mi></mrow></msub></math>$ is injective, and its thickening is a Helly graph.

Concerning Artin groups of Euclidean types $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mrow><mi>Ã</mi></mrow><mrow><mi>n</mi></mrow></msub></math>$ and $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mrow><mover accent="true"><mrow><mi>C</mi></mrow><mo accent="true">~</mo></mover></mrow><mrow><mi>n</mi></mrow></msub></math>$ , we show that the natural piecewise $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msup><mrow><mi>ℓ</mi></mrow><mrow><mi>∞</mi></mrow></msup></math>$ 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 $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>K</mi><mo class="MathClass-open" stretchy="false">(</mo><mi>π</mi><mo class="MathClass-punc">,</mo><mn>1</mn><mo class="MathClass-close" stretchy="false">)</mo></math>$ 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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