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

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 $SL(2,\mathbb{C})$. 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.