Generating pairs for SL(n, Z)

It is well known that for all n≥3, the group SL(n,Z) has a finite presentation given by its n2−n transvections, subject to the Steinberg relations. Also by a 1962 theorem of Trott, if n is odd then 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 SL(n,Z) is generated by two elements of orders 2 and 3 precisely when n≥5. On the other hand, little appears to be known about 2-generator presentations for SL(n,Z) for n≥3. In this paper, some finite 2-generator presentations are given for SL(3,Z), which as far as the authors are aware, are the only 2-generator finite presentations known for SL(3,Z). Also some new generating pairs are given for SL(n,Z) for n≥3. In particular, some of these extend Trott's 1962 theorem by showing that SL(n,Z) is generated by two elements, one of order 2 and the other of infinite order, for all n>2.