R. Thompson introduced three groups $F\subset T\subset V$. The question concerning amenability of $F$ is still unanswered and has attracted much attention. I have read that Thompson group $V$ contains a copy of the free group $F\_2$ (with two generators), in particular it is not amenable. Does anyone know an explicit embedding or can give me a reference for it? Thank you very much for the help.



Explicit free subgroup in Thompson's group $V$
Thompson's group $T$ of the circle itself contains a copy of $F\_2$: this is indeed easy to find a [ping-pong pair](https://en.wikipedia.org/wiki/Ping-pong_lemma) starting from 4 disjoint intervals. That $T$ is non-amenable is even easier, because it does not preserve any probability on Borel subsets on the circle (immediate once we check that the only probabilities on Borel subsets of the interval $[0,1]$ invariant by $F$, are supported by $\{0,1\}$, and hence after identifying $0=1$ the only $F$-invariant probability on the circle is the Dirac at $0=1$, which of course is not $T$-invariant).


