The minimal index of a torsion-free subgroup of SL3(Z)
Year: 2026
Type: preprint
Abstract: We prove that the minimal index of a torsion-free subgroup of finite index in SL3(ℤ) is 192. This answers, for N = 3, a question of Nahm recorded by Soulé, who asked for the minimal index of a torsion-free subgroup of finite index in SLN(ℤ), N ≥ 3. The minimum is eight times the classical divisibility bound 24 given by the finite subgroups. The minimum is attained by a congruence subgroup of level 4 whose image modulo 2 is generated by a Singer cycle. For the lower bound, the congruence subgroup property reduces the problem to open subgroups of ∏p SL3(ℤp); the odd primes are eliminated by index and Sylow arguments, and the remaining cases are excluded by a 2-adic Frattini analysis of the first congruence layers, which uses a short exhaustive computation in the 8-dimensional 𝔽2-space 𝔰𝔩3(𝔽2). MSC 2020: Primary 20H05; Secondary 20E07, 20G25, 11F06. The record contains the paper, its LaTeX source, and Python programs with recorded outputs that verify the finite computations used in the paper.
Author: Na, Sungsoo
Language: English
Subfield: Algebra and Number Theory
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7225570201