The second integral homology of \rm SL₂(ℤ[1/n])

2025/03/15 by Mirzaii, Behrooz, Ramos, Bruno Reis, Verissimo, Thiago
#19G99 19G99 #20G10 #FOS: Mathematics #K-Theory and Homology (math.KT)

paper · doi:10.48550/arxiv.2503.12190

Abstract

In this article, we explore the second integral homology, or Schur multiplier, of the special linear group \rm SL₂(ℤ[1/n]) for a positive integer n. We definitively calculate the group structure of H₂(\rm SL₂(ℤ[1/n]),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 H₂(\rm SL₂(ℤ[1/n]),ℤ) when n is not divisible by any of those specific primes.

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