# Thompson group $V$

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.

• $T$ itself contains a copy of $F_2$: this is indeed easy to find a ping-pong pair 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 invariant by $F$ are supported by $\{0,1\}$). – YCor Mar 22 '16 at 21:20
• Thank you for the answer. Just one thing, what do you mean by "ping-pong pair"? Thank you for the help. – John N. Mar 22 '16 at 21:25
• Given a group with an action on a set, it is a pair satisfying the ping-pong lemma. That is, in the language of en.wikipedia.org/wiki/Ping-pong_lemma (and the space being the circle), it is a pair $(a_1,a_2)$ as in the ping-pong lemma for cyclic subgroups [for future readers: I refer to today's 2016/03/22 version, which can be found in the history in case it doesn't fit]. – YCor Mar 22 '16 at 21:47
• Thank you very much. If you copy these comments in an answer I can vote for your answer – John N. Mar 22 '16 at 21:50

Thompson's group $T$ of the circle itself contains a copy of $F_2$: this is indeed easy to find a ping-pong pair 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).

If $\kappa$, $\lambda$, $\mu$ and $\nu$ are generators of Thompson's Group $V$ which satisfy the defining relations given on Page 50 in

Graham Higman, Finitely presented infinite simple groups, Notes on Pure Mathematics, Department of Pure Mathematics, Australian National University, Canberra, 1974. MR0376874 (51 #13049)

an explicit embedding of the free group of rank $2$ is given by $$\varphi: \ {\rm F}_2 = \langle a, b \rangle \ \rightarrow \ V, \ \ a \mapsto (\kappa \mu)^2, \ b \mapsto \lambda(\kappa \nu)^2 \lambda \kappa.$$ This can be found with GAP as follows:

gap> LoadPackage("rcwa");
gap> k := ClassTransposition(0,2,1,2);; l := ClassTransposition(1,2,2,4);;
gap> m := ClassTransposition(0,2,1,4);; n := ClassTransposition(1,4,2,4);;
gap> V := Group(k,l,m,n); # Thompson's group V with generators as above
<(0(2),1(2)),(1(2),2(4)),(0(2),1(4)),(1(4),2(4))>
gap> F2 := FreeGroup("a","b");;
gap> phi := IsomorphismRcwaGroup(F2);
[ a, b ] -> [ <wild rcwa permutation of Z with modulus 8>,
<wild rcwa permutation of Z with modulus 8> ]
gap> IsSubgroup(V,Image(phi)); # we are lucky to have an embedding into V
true
gap> F4 := FreeGroup("k","l","m","n");;
gap> psi := EpimorphismByGenerators(F4,V);
[ k, l, m, n ] -> [ ( 0(2), 1(2) ), ( 1(2), 2(4) ), ( 0(2), 1(4) ),
( 1(4), 2(4) ) ]
gap> a := PreImagesRepresentative(psi,Image(phi,F2.1));
(k*m)^2
gap> b := PreImagesRepresentative(psi,Image(phi,F2.2));
l*(k*n)^2*l*k