Uniform two-generator presentations for $SL_n(\mathbb{Z})$ with polynomial complexity bounds

Abstract

We give a uniform explicit construction of finite two-generator presentations for the special linear groups over the integers in all ranks at least three. The construction builds on the generating-pair work of Conder--Liversidge--Vsemirnov and on a standard Tietze-elimination observation pointed out by Button. It recovers Trott's odd-rank generating pair and extends the same monomial/transvection form uniformly to even rank by a sign correction. After rebalancing, the construction has quadratic transvection words, quartically many relators, and sextic total relator length. We also derive several consequences, including infinite--infinite and finite--finite variants, consequences for congruence quotients, a presentation for the projective quotient, and an exact relator count, valid for both the unbalanced and balanced presentations.