Local-to-Global-rigidity of lattices in $S{L}_n\left(𝕂\right)$

Escalier, Amandine

Annales de l'Institut Fourier, Tome 72 (2022) no. 5, pp. 1733-1771

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 $PS{L}_n\left(𝕂\right)$ with $n\ge 4$ 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 $S{L}_n\left(𝕂\right)$.

The proof is the opportunity to prove a result on the local structure of the building. We show that if we fix a $PS{L}_n\left(𝕂\right)$-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 $PS{L}_n\left(𝕂\right)$ lorsque $n\ge 4$ 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 $S{L}_n\left(𝕂\right)$.

La démonstration est l’occasion de prouver un résultat sur la structure locale des immeubles. Nous montrons que si l’on fixe une $PS{L}_n\left(𝕂\right)$-orbite dans l’immeuble, alors un sommet est uniquement déterminé par les sommets voisins contenus dans cette orbite.

Classification : 20F65
Keywords: Lattices, Buildings, Rigidity, Local Field
Mots-clés : Réseaux, Immmeubles, Régidité, Corps locaux

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