The second homology of SL_2 of S-integers

TL;DR

This paper computes the second homology groups of SL_2 over certain S-integers, constructs explicit generators, and explores their relation to universal central extensions and K_2 symbols.

Contribution

It provides explicit descriptions of H_2(SL_2(Z[1/m])) for specific m, and establishes the universal central extension property using these results.

Findings

Explicit generators for H_2(SL_2(Z[1/m])) when m is divisible by 6.

The projection St(2, Z[1/m]) --> SL_2(Z[1/m]) is a universal central extension under certain conditions.

Constructed homology classes correspond to symbols in K_2 of the ring.

Abstract

We calculate the structure of the finitely-generated groups H_2(SL_2(Z[1/m])) when m is a multiple of 6. We construct explicit homology classes which generate these groups and have prescribed orders. When n is at least 2 and m is the product of the first n primes, we combine our results with those of Jun Morita to deduce that the projection St(2, Z[1/m]) --> SL_2(Z[1/m]) is a universal central extension, where St(2,-) is the rank one Steinberg group. The main structure theorem applies more generally to rings of S-integers with sufficiently many units. For a wide class of rings, the explicit homology classes we construct map to symbols in rank one K_2 of the ring.