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

TL;DR

This paper investigates the growth of torsion in the cohomology of arithmetic hyperbolic 3-manifolds and relates it to the properties of the twisted Ruelle zeta function, revealing exponential growth patterns.

Contribution

It establishes the exponential growth of 2-torsion in the cohomology and links it to the leading coefficient of the twisted Ruelle zeta function's Laurent expansion.

Findings

2-torsion in cohomology grows exponentially

Leading coefficient of Ruelle zeta function relates to torsion orders

Provides new insights into the structure 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,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.