On the cohomology of GL_2 and SL_2 over imaginary quadratic fields

TL;DR

This paper presents extensive computations of the cohomology groups of GL_2 and SL_2 over imaginary quadratic integer rings, revealing exponential growth in torsion and extending previous results using Voronoi complex techniques.

Contribution

It advances the understanding of cohomology of these groups over imaginary quadratic fields by providing new computational data beyond prior studies.

Findings

Cohomology computations extend previous results significantly.

Exponential growth observed in torsion subgroup of H^2 as discriminant increases.

Data aligns with bounds proposed by Rohlfs.

Abstract

We report on computations of the cohomology of GL_2(O_D) and SL_2(O_D), where D<0 is a fundamental discriminant. These computations go well beyond earlier results of Vogtmann and Scheutzow. We use the technique of homology of Voronoi complexes, and our computations recover the integral cohomology away from the primes 2, 3. We observed exponential growth in the torsion subgroup of H^2 as $D$ increases, and compared our data to bounds of Rohlfs.