The abelianization of SL2(Z[1/m])

For all m > 1, we prove that the abelianization of SL2(Z[1/m]) is (1) trivial if 6 |m; (2) Z/3Z if 2 |m m and gcd(3, , m ) = 1; (3) Z / 4 Z if 3 |m m and gcd(2, m ) = 1; and (4) Z / 12 Z similar to= Z / 3 Z x Z/4Z if gcd(6, , m ) = 1. This completes known computational results of Bui Anh & Ellis for m < 50. The proof is elementary, and does not use the congruence subgroup property. We also find a new presentation for SL2(Z[1/2]) with two generators and three relators. This also gives new, simple presentations for the finite groups SL2(Z/mZ), where m is odd. (c) 2024 Elsevier Inc. All rights are reserved, including those for text and data mining, AI training, and similar technologies.