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 generalized identities in terms of torsion and exponent properties of the unit group of the underlying ring. As applications, we construct new families of infinite finitely generated solvable groups satisfying generalized identities of prescribed lengths and provide a counterexample to a question of Sherman.