Generating pairs for SL(n, Z)

Abstract

It is well known that for all $n\ge 3$, the group $\mathrm{SL}(n,Z)$ has a finite presentation given by its $n^2-n$ transvections, subject to the Steinberg relations. Also by a 1962 theorem of Trott, if n is odd then $\mathrm{SL}(n,Z)$ is generated by two elements, one of infinite order, and by the combined work of Tamburini, J.S. Wilson and Vsemirnov and others (from 1993 to 2021), it is now known that $\mathrm{SL}(n,Z)$ is generated by two elements of orders 2 and 3 precisely when $n\ge 5$. On the other hand, little appears to be known about 2-generator presentations for $\mathrm{SL}(n,Z)$ for $n\ge 3$. In this paper, some finite 2-generator presentations are given for $\mathrm{SL}(3,Z)$, which as far as the authors are aware, are the only 2-generator finite presentations known for $\mathrm{SL}(3,Z)$. Also some new generating pairs are given for $\mathrm{SL}(n,Z)$ for $n\ge 3$. In particular, some of these extend Trott's 1962 theorem by showing that $\mathrm{SL}(n,Z)$ is generated by two elements, one of order 2 and the other of infinite order, for all $n>2$.