Groups of fast homeomorphisms of the interval and the ping-pong argument

Collin Bleak; Matthew G. Brin; Martin Kassabov; Justin Tatch Moore; Matthew C. B. Zaremsky

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arXiv · 1701.08321

Groups of fast homeomorphisms of the interval and the ping-pong argument

Collin Bleak·Matthew G. Brin·Martin Kassabov·Justin Tatch Moore·Matthew C. B. Zaremsky

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Abstract

We adapt the Ping-Pong Lemma, which historically was used to study free products of groups, to the setting of the homeomorphism group of the unit interval. As a consequence, we isolate a large class of generating sets for subgroups of $\mathrm{Homeo}_+(I)$ for which certain finite dynamical data can be used to determine the marked isomorphism type of the groups which they generate. As a corollary, we will obtain a criteria for embedding subgroups of $\mathrm{Homeo}_+(I)$ into Richard Thompson's group $F$. In particular, every member of our class of generating sets generates a group which embeds into $F$ and in particular is not a free product. An analogous abstract theory is also developed for groups of permutations of an infinite set.

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BibTeX RIS

Collin Bleak, Matthew G. Brin, Martin Kassabov, Justin Tatch Moore, Matthew C. B. Zaremsky. 2017-01-28. Groups of fast homeomorphisms of the interval and the ping-pong argument. https://arxiv.org/abs/1701.08321

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