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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)



Free products in R. Thompson's group $ V$

Authors: Collin Bleak and Olga Salazar-Díaz
Journal: Trans. Amer. Math. Soc. 365 (2013), 5967-5997
MSC (2010): Primary 20F65, 37C85, 20E07, 20E32
Published electronically: June 19, 2013
MathSciNet review: 3091272
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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 $ \mathfrak{C}$. We draw attention to the class $ \mathcal {D}_{(V,\mathfrak{C})}$ of groups which have embeddings as demonstrative subgroups of $ V$ whose class can be used to assist in forming various products. Note that $ \mathcal {D}_{(V,\mathfrak{C})}$ contains all finite groups, the free group on two generators, and $ \mathbf {Q}/\mathbf {Z}$, and is closed under passing to subgroups and under taking direct products of any member by any finite member. If $ G\leq V$ and $ H\in \mathcal {D}_{(V,\mathfrak{C})}$, then $ G\wr H$ embeds into $ V$. Finally, if $ G$, $ H\in \mathcal {D}_{(V,\mathfrak{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.

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Additional Information

Collin Bleak
Affiliation: School of Mathematics and Statistics, Mathematical Institute, University of St. Andrews, North Haugh, St. Andrews, Fife KY16 9SS, Scotland

Olga Salazar-Díaz
Affiliation: Escuela de Matemáticas, Universidad Nacional de Colombia, Medellín, Colombia

Keywords: Thompson's group $V$, subgroup structure, group actions, demonstrative groups, $co\mathscr{CF}$ groups
Received by editor(s): November 13, 2009
Received by editor(s) in revised form: February 22, 2011, and March 7, 2012
Published electronically: June 19, 2013
Article copyright: © Copyright 2013 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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