Local-to-Global-rigidity of lattices in SL n (đť•‚)
Annales de l'Institut Fourier, Online first, 39 p.

A vertex-transitive graph 𝒢 is called Local-to-Global rigid if there exists R>0 such that every other graph whose balls of radius R are isometric to the balls of radius R in 𝒢 is covered by 𝒢. An example of such a graph is given by the Bruhat–Tits building of PSL n (𝕂) with n≥4 and 𝕂 a non-Archimedean local field of characteristic zero. In this paper we extend this rigidity property to a class of graphs quasi-isometric to the building including torsion-free lattices of SL n (𝕂).

The proof is the opportunity to prove a result on the local structure of the building. We show that if we fix a PSL n (đť•‚)-orbit in it, then a vertex is uniquely determined by the neighbouring vertices in this orbit.

Un graphe transitif 𝒢 est dit Local-Global rigide s’il existe R>0 tel que tout autre graphe dont les boules de rayon R sont isométriques aux boules de rayon R de 𝒢 est revêtu par 𝒢. Un exemple de tel graphe est donné par l’immeuble de Bruhat–Tits de PSL n (𝕂) lorsque n≥4 et 𝕂 est un corps local non-Archimédien de caractéristique nulle. Dans cet article nous étendons cette propriété de rigidité à une classe de graphes quasi-isométriques à l’immeuble, incluant les réseaux sans torsion de SL n (𝕂).

La démonstration est l’occasion de prouver un résultat sur la structure locale des immeubles. Nous montrons que si l’on fixe une PSL n (𝕂)-orbite dans l’immeuble, alors un sommet est uniquement déterminé par les sommets voisins contenus dans cette orbite.

Received:
Revised:
Accepted:
Online First:
DOI: 10.5802/aif.3490
Classification: 20F65
Keywords: Lattices, Buildings, Rigidity, Local Field
Escalier, Amandine 1

1 Université Paris Cité and Sorbonne Université, CNRS, IMJ-PRG, F-75013 Paris (France)
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Escalier, Amandine. Local-to-Global-rigidity of lattices in $SL_n(\protect \mathbb{K})$. Annales de l'Institut Fourier, Online first, 39 p.

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