Twisted limit formula for torsion and cyclic base change

Bergeron, Nicolas; Lipnowski, Michael

Journal de l’École polytechnique — Mathématiques, Tome 4 (2017), pp. 435-471.

Soit $G$ le groupe des points complexes d’un groupe de Lie semi-simple réel dont le rang fondamental est égal à 1, par exemple $G={\mathrm{SL}}_2\left(ℂ\right)\times {\mathrm{SL}}_2\left(ℂ\right)$ ou ${\mathrm{SL}}_3\left(ℂ\right)$. Alors le rang fondamental de $G$ est égal à $2$ et, selon la conjecture faite dans [3], les réseaux dans $G$ devraient avoir « peu » — dans le sens très faible de « sous-exponentiel en le co-volume » — de torsion homologique. En utilisant le changement de base, nous exhibons des suites de réseaux le long desquelles la torsion homologique croît exponentiellement avec la racine carrée du volume. Ce comportement est déduit d’un théorème général qui compare les torsions $L^2$ tordues et non tordues dans la situation générale d’un changement de base. Nous utilisons également une version équivariante précise du « Théorème de Cheeger-Müller » démontrée par le second auteur [23].

Let $G$ be the group of complex points of a real semi-simple Lie group whose fundamental rank is equal to $1$, e.g. $G={\mathrm{SL}}_2\left(ℂ\right)\times {\mathrm{SL}}_2\left(ℂ\right)$ or ${\mathrm{SL}}_3\left(ℂ\right)$. Then the fundamental rank of $G$ is $2$, and according to the conjecture made in [3], lattices in $G$ should have ‘little’ — in the very weak sense of ‘subexponential in the co-volume’ — torsion homology. Using base change, we exhibit sequences of lattices where the torsion homology grows exponentially with the square root of the volume. This is deduced from a general theorem that compares twisted and untwisted $L^2$-torsions in the general base-change situation. This also makes uses of a precise equivariant ‘Cheeger-Müller Theorem’ proved by the second author [23].

Classification : 11F75, 11F70, 11F72, 58J52
Keywords: Homological torsion, limit multiplicities, base change
Mot clés : Torsion homologique, multiplicité limite, changement de base

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