Generalized torsion in triangular matrix groups

Abstract

An element of a group is called generalized torsion if a finite product of its conjugates equals the identity. We characterize the generalized torsion subset of upper triangular matrix groups over commutative rings and determine when these groups satisfy positive generalized identities in terms of torsion and exponent properties of the unit group of the underlying ring. Moreover, as an application of the developed theory, we construct triangular matrix groups satisfying positive generalized identities of prescribed degrees and obtain lower bounds for the minimum degrees of such identities. We also provide a counterexample to a question of Sherman concerning groups generated by elements of infinite order.