## Abstract

We investigate some product structures in R. Thompson's group *V*, primarily by studying the topological dynamics associated with *V*'s action on the Cantor set C. We draw attention to the class D_{(V,C)} of groups which have embeddings as demonstrative subgroups of V whose class can be used to assist in forming various products. Note that D_{(V,C)} contains all finite groups, the free group on two generators, and Q/Z, and is closed under passing to subgroups and under taking direct products of any member by any finite member. If G≤*V *and H ∈ D_{(V,C)}, then G~H embeds into *V*. Finally, if G, H ∈ D_{(V,C)}, then G*H embeds in *V*.

Using a dynamical approach, we also show the perhaps surprising result that Z^{2} * Z does not embed in *V*, even though *V *has many embedded copies of Z^{2 }and has many embedded copies of free products of various pairs of its subgroups.

Original language | English |
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Pages (from-to) | 5967-5997 |

Number of pages | 31 |

Journal | Transactions of the American Mathematical Society |

Volume | 365 |

Issue number | 11 |

Early online date | 19 Jun 2013 |

DOIs | |

Publication status | Published - 1 Nov 2013 |

## Keywords

- R. Thompson Groups
- Homeomorphisms
- Cantor set

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