The second integral homology of ${\mathrm{S}\mathrm{L}}_2(\mathbb{Z}[1\mathrm{/}n])$

In this article, we explore the second integral homology, or Schur multiplier, of the special linear group ${\mathrm{S}\mathrm{L}}_2(\mathbb{Z}[1\mathrm{/}n])$ for a positive integer $n$. We definitively calculate the group structure of $H_2({\mathrm{S}\mathrm{L}}_2(\mathbb{Z}[1\mathrm{/}n]),\mathbb{Z})$ 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 $H_2({\mathrm{S}\mathrm{L}}_2(\mathbb{Z}[1\mathrm{/}n]),\mathbb{Z})$ when $n$ is not divisible by any of those specific primes.