A nonstandard construction of direct limit group actions

Abstract

Manevitz and Weinberger (1996) proved that the existence of effective $K$-Lipschitz $\mathbb{Z}/n\mathbb{Z}$-actions implies the existence of effective $K$-Lipschitz $\mathbb{Q}/\mathbb{Z}$-actions for all compact connected manifolds with metrics, where $K$ is a fixed Lipschitz constant. The $\mathbb{Q}/\mathbb{Z}$-actions were constructed from suitable actions of a sufficiently large hyperfinite cyclic group ${}^{\ast}{\mathbb{Z}}/\gamma{}^{\ast}{\mathbb{Z}}$ in the sense of nonstandard analysis. By modifying their construction, we prove that for every direct system $\left(\Lambda,G_{\lambda},i_{\lambda\mu}\right)$ of torsion groups with monomorphisms, the existence of effective $K$-Lipschitz $G_{\lambda}$-actions implies the existence of effective $K$-Lipschitz $\varinjlim G_{\lambda}$-actions. This generalises Manevitz and Weinberger's result.