Analytic and Reidemeister torsion for non-compact locally symmetric spaces
Final Report Abstract
In this research project, several aspects related to the analytic torsion and its applications on non compact locally symmetric spaces were studied. The first central step in dealing with the analytic torsion on non compact locally symmetric spaces is to show that a regularized heat trace can be defined and that it has a suitable short time aysmptotic expansion. While this problem is solved in the real rank one case, it has been open for higher rank locally symmetric spaces. The geometry of such spaces at their non compact parts is much more complicated than in the rank one case, since they only compactify to manifolds with corners. This leads in particular to the difficulty that the ’ends’ of the locally symmetric space cannot be separated any more. In our first project, in joint work with Rafe Mazzeo we proved a full asymptotic expansion for the heat kernel of the Laplacian on the higher rank locally symmetric space X = Γ\ SL3 (R)/ SO(3), where Γ is a finite index subgroup of SL3 (Z). The main tool was to adapt the methods developed by Richard Melrose for the construction of the heat kernel on manifolds with cylindrical ends to the present setting. More precisely, we constructed a suitable larger and less singular space which is related to the original space X, or, more precisely, to X × X × (0, ∞), via a sequence of blow-ups. On that space, the Laplacian on the preimages of the non compact parts could be described by simpler model operators and thus the heat equation could be solved explicitly up to infinite order. Our methods should generalize to any higher rank locally symmetric space and to other geometric differential operators. We also hope that they could be useful to treat other problems in geometric analysis on higher rank locally symmetric spaces in which the heat kernel is relevant. In a second project, in joint work with Jean Raimbault we studied the growth of cohomological torsion for congruence subgroups of Bianchi groups [PR]. We proved that for a fixed neat principal congruence subgroup of a Bianchi group the order of the torsion part of its 2nd cohomology group with coefficients in an integral lattice associated to the m-th symmetric power of the standard representation of SL2 (C) grows exponentially in m2 . We also established a limit multiplicity formula for combinatorial torsion for sequences of higher dimensional non compact arithmetic hyperbolic manifolds. The main tools were a gluing formula that related the analytic torsion to the combinatorial Reidemeister torsion [Pf] as well as an investigation of the regulator that appears as a defect in the formula relating Reidemeister- to cohomological torsion.
Publications
- A gluing formula for the analytic torsion on hyperbolic manifolds with cusps, Journal of the Institute of Mathematics of Jussieu
J. Pfaff
(See online at https://dx.doi.org/10.1017/S1474748015000237) - On the torsion in symmetric powers on congruence subgroups of Bianchi groups
J. Pfaff, J. Raimbault
