On the centralizers of finite subgroups in quasi-HNN groups

Quasi-HNN groups can be characterized as a generalization of HNN groups in the sense that there is a stable letter $t$ such that the element $t^2$ is in the base. In this paper we show that under certain conditions the centralizer of a finite subgroup in a quasi-HNN group is contained in a conjugate of the base. As an application we show that the centralizer of a finite subgroup in a one-relator group is contained in a conjugate of a one-relator subgroup of shorter relator.