Twisted limit formula for torsion and cyclic base change

Nicolas Bergeron,Michael Lipnowski

doi:10.5802/jep.47

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. $G= \SL_2 (\C) \times \SL_2 (\C)$ or $\SL_3 (\C)$. Then the fundamental rank of $G$ is $2,$ and according to the conjecture made in \cite{BV}, 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 \emph{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-Muller Theorem' proved by the second author \cite{Lip1}.

Highlights

If b(j2) = 0 for some j, it follows that cohomology is abundant
We prove Proposition 1.2 for certain families {Γn} in Section 4.6 but our real interest here is rather how the torsion cohomology grows
The authors hope that the limit multiplicity formula (1.6.1) together with the twisted endoscopic comparison implicit in Section 7 will be of interest independent of torsion in cohomology

Summary

Introduction

Let Γn ⊂ Γ be a sequence of finite index, σ-stable subgroups of G It follows from the general Proposition 1.2 below that bj(Γn) vol(Γσn\SL2(R)). The proof of (1.5.1) crucially uses the equivariant Cheeger-Müller theorem, proven by Bismut-Zhang [5] This enables us to compute the left side of (1.5.1) (up to a controlled integer multiple of log 2) by studying the eigenspaces of the Laplace operators of the metrized local system associated to ρ together with their σ action. The authors hope that the limit multiplicity formula (1.6.1) together with the twisted endoscopic comparison implicit in Section 7 will be of interest independent of torsion in cohomology These computations complement work by Borel-LabesseSchwermer [6] and Rohlfs-Speh [30]. They would like to thank an anonymous referee for pointing out an error in the first version of this paper

The simple twisted trace formula

Lefschetz number and twisted analytic torsion

Computations on a product

General base change

Generalities on Reidemeister torsion and the Cheeger-Müller theorem