Generic groups acting on regular trees

Abért, Miklós; Glasner, Yair

doi:10.1090/S0002-9947-09-04662-5

Generic groups acting on regular trees
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by Miklós Abért and Yair Glasner PDF

Trans. Amer. Math. Soc. 361 (2009), 3597-3610 Request permission

Abstract:

Let $T$ be a $k$-regular tree ($k\geq 3$) and $A=\mathrm {Aut}(T)$ its automorphism group. We analyze a generic finitely generated subgroup $\Gamma$ of $A$. We show that $\Gamma$ is free and establish a trichotomy on the closure $\overline {\Gamma }$ of $\Gamma$ in $A.$ It turns out that $\overline {\Gamma }$ is either discrete, compact or has index at most $2$ in $A$.

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