THE TORSION IN SYMMETRIC POWERS ON CONGRUENCE SUBGROUPS OF BIANCHI GROUPS

Torsion à coefficients dans les puissances symétriques pour les groupes de Bianchi

Jonathan Pfaff , Jean Raimbault (1)

1 IMT - Institut de Mathématiques de Toulouse UMR5219

In this paper we prove that for a fixed neat principal congruence subgroup of a Bianchi group the order of the torsion part of its second cohomology group with coefficients in an integral lattice associated to the m-th symmetric power of the standard representation of SL 2 (C) grows exponentially in m 2. We give upper and lower bounds for the growth rate. Our result extends a a result of W. Müller and S. Marshall, who proved the corresponding statement for closed arithmetic 3-manifolds, to the finite-volume case. We also prove a limit multiplicity formula for combinatorial Reidemeister torsions on higher dimensional hyperbolic manifolds.

Domaines: Géométrie différentielle [math.DG]; Théorie des nombres [math.NT]; Topologie géométrique [math.GT]

Jonathan Pfaff, Jean Raimbault. THE TORSION IN SYMMETRIC POWERS ON CONGRUENCE SUBGROUPS OF BIANCHI GROUPS. Transactions of the American Mathematical Society, 2020, 373 (1), pp.109-148. ⟨10.1090/tran/7875⟩. ⟨hal-02358132⟩