The second integral homology of ${\rm SL}_2(\mathbb{Z}[1/n])$
Behrooz Mirzaii, Bruno Reis Ramos, Thiago Verissimo
TL;DR
The paper determines the second integral homology $H_2(\mathrm{SL}_2(\mathbb{Z}[1/n]),\mathbb{Z})$ for square-free $n$ with a prime factor in $\{2,3,5,7,13\}$ by combining the amalgamated-product structure of $\mathrm{SL}_2(\mathbb{Z}[1/n])$, Mayer–Vietoris sequences, and careful analysis of $H_1$ and $H_2$ for subgroups $\Gamma_0(n,p)$. It provides explicit decompositions in terms of $\mathbb{Z}$ and cyclic factors $\mathbb{Z}/(p-1)$, as well as exact sequences that describe the dependence on the primes dividing $n$ (via $r_p$ and ${\rm scpd}(n,2730)$). The work extends Adem–Naffah, Hutchinson, and Bui–Ellis results to general $n$ with the specified prime divisors, and it establishes surjectivity results for the maps from $H_2(\Gamma(n,p),\mathbb{Z})$ into $H_2(\mathrm{SL}_2(\mathbb{Z}[1/n]),\mathbb{Z})$. It also offers a rank conjecture and a conjectural description for $H_2$ when $n$ avoids those primes, highlighting connections to algebraic $K$-theory via $K_2(\mathbb{Q})$ and tame symbols.
Abstract
In this article, we explore the second integral homology, or Schur multiplier, of the special linear group ${\rm SL}_2(\mathbb{Z}[1/n])$ for a positive integer $n$. We definitively calculate the group structure of $H_2({\rm SL}_2(\mathbb{Z}[1/n]),\mathbb{Z})$ when $n$ is divisible by one of the primes $2$, $3$, $5$, $7$ or $13$. For a general $n > 1$, we offer a partial description by placing the homology group within an exact sequence, and we investigate its rank. Finally, we propose a conjectural structure for $H_2({\rm SL}_2(\mathbb{Z}[1/n]),\mathbb{Z})$ when $n$ is not divisible by any of those specific primes.