### 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 $\prescript{\ast}{}{\mathbb{Z}}/\gamma\prescript{\ast}{}{\mathbb{Z}}$ 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.

#### Keywords

goup actions; direct limits of groups; locally finite groups; nonstandard analysis

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PDFISSN: 1331-0623 (Print), 1848-8013 (Online)