Infinite-dimensional cohomology of SL_2(Z[t, 1/t])

TL;DR

This paper constructs a map from the classifying space of SL_2 over a Laurent polynomial ring to a Euclidean building, demonstrating that the second cohomology group of this group is infinite-dimensional.

Contribution

It introduces a novel method to analyze the cohomology of SL_2 over Laurent polynomial rings by linking classifying spaces to Euclidean buildings and constructing infinite cocycles.

Findings

H2(SL_2(J[t,1/t]);F) is infinite-dimensional

Constructed an explicit family of independent cocycles

Mapped the 3-skeleton of the classifying space to a Euclidean building

Abstract

For J an integral domain and F its field of fractions, we construct a map from the 3-skeleton of the classifying space for {\Gamma} = SL_2(J[t,1/t]) to a Euclidean building on which {\Gamma} acts. We then find an infinite family of independent cocycles in the building and lift them to the classifying space, thus proving that the cohomology group H2(SL_2(J[t,1/t]);F) is infinite-dimensional.