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Nicolas Bergeron; Michael Lipnowski
Twisted limit formula for torsion and cyclic base change
(Formule de multiplicité limite tordue pour la torsion et changement de base cyclique)
Journal de l'École polytechnique — Mathématiques, 4 (2017), p. 435-471, doi: 10.5802/jep.47
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Class. Math.: 11F75, 11F70, 11F72, 58J52
Keywords: Homological torsion, limit multiplicities, base change

Résumé - Abstract

Let $G$ be the group of complex points of a real semi-simple Lie group whose fundamental rank is equal to $1$, e.g. $G= \mathrm{SL}_2 (\mathbb{C}) \times \mathrm{SL}_2 (\mathbb{C})$ or $\mathrm{SL}_3 (\mathbb{C})$. 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].


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