Analytic torsion and $L^2$-torsion of compact locally symmetric manifolds

Abstract

In this paper we define a regularized version of the analytic torsion for arithmetic quotients of a symmetric space of non-positive curvature. The definition is based on the study of the renormalized trace of the corresponding heat operators, which is defined as the geometric side of the Arthur trace formula applied to the heat operator.

On the growth of torsion in the cohomology of arithmetic groups

“…(2) X ′ (ρ 0 (m)) be the L 2 -torsion with respect to ρ 0 (m) (see [MP,section 5]). If we apply [MP,Proposition 1.2] to the irreducible components of ρ 0 (m), it follows that there exists c > 0 such that…”

“…Using the definition of ρ 0 (m) by (3.26), [MP,(5.21)] and [MP,Proposition 5.3], it follows that log T…”

“…q is as in (1.3), then by[MP, Proposition 6.7] one has log T(2)X ′ (τ (m)) = log T (2) X ′ (τ (m) θ ) = C p,q vol(X ′ )m dim τ (m) + O(dim(τ (m)),as m → ∞. Thus putting everything together we obtain log T X ′ (ρ(m)) = C p,q vol(X ′ )m dim(ρ(m)) + O(dim(ρ(m))),…”

A Gluing Formula for the Analytic Torsion on Hyperbolic Manifolds With Cusps

“…We also remark that at least in the compact locally symmetric case the analytic torsion with coefficients in a flat bundle induced by a finite-dimensional representation of G is always equal to 1 if the dimension of the manifold is even, [MS], [BMZ], [MP3]. Moreover, the present odd-dimensional hyperbolic case is exactly the case corresponding to an irreducible symmetric space of real rank one which is odd dimesional.…”

“…Now, although the (regularized) analytic torsion can in general not be computed explicitly for a fixed locally symmetric space and strongly acyclic coefficient systems, its asymptotic behaviour can be determined explicitly if either the manifold or the local system varies. In the compact case, results about the asymptotic behaviour of the analytic torsion were established by Bergeron and Venkatesh [BV], by Bismut, Ma and Zhang [BMZ] and by Müller and the author [MP1], [MP3]. The extension of these results to the non-compact, finite volume real rank one case is due to Raimbault [Ra] and to Müller and the author [MP2], [MP5].…”