On the torsion in the cohomology of arithmetic hyperbolic 3-manifolds

Abstract

Abstract. In this paper we consider the cohomology of a closed arithmetic hyperbolic 3-manifold with coefficients in the local system defined by the even symmetric powers of the standard representation of SL(2, C). The cohomology is defined over the integers and is a finite abelian group. We show that the order of the 2nd cohomology grows exponentially as the local system grows. We also consider the twisted Ruelle zeta function of a closed arithmetic hyperbolic 3-manifold and we express the leading coefficient of its Laurent expansion at the origin in terms of the orders of the torsion subgroups of the cohomology.

The torsion in symmetric powers on congruence subgroups of Bianchi groups

“…In this paper, we study the twisted cohomological torsion quantitatively for a fixed principal congruence subgroup of a Bianchi group under a variation of the local system. Bianchi groups represent all classes of non-uniform lattices in SL 2 (C) ; thus our result complements the study of this question for arithmetic lattices in SL 2 (C) defined over imaginary quadratic fields done by Simon Marshall and Werner Müller in [MM13] (where the authors give an equality for the asymptotic torsion size, while we only get upper and lower bounds for the growth rate).…”

Growth of homology torsion in finite coverings and hyperbolic volume

“…For related results see e.g. [1,4,5,9,23,24,25]. Probably C. Gordon [11] was the first to ask about the asymptotic growth of torsion of homology in finite, albeit abelian, coverings.…”

On the growth of torsion in the cohomology of arithmetic groups

“…Let τ X (ρ R ) be the Reidemeister torsion of the complex C * (L; V R ) with respect to volume elements defined by these bases. Then it follows from (2.11) and (2.12) as in[BV, section 2.2] that (2.13) log τ X (ρ R ) = d q=0 (−1) q+1 log |H q (Γ, M)|.See also[MaMü, Proposition 2.3] and [Tu, Lemma 2.1.1]. Since the coboundary operators of the complexes C * (L; V R ) and C * (L; V ), respectively, are induced by the coboundary operators of C * (L; M), it follows from the definition of the Reidemeister torsion that τ X (ρ R ) = τ X (ρ).…”