# Generic groups acting on regular trees Academic Article

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### abstract

• 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 $A$.

### publication date

• January 1, 2009