What are examples of expander family of 3-transitive Schreier graphs? 
Meaning for an action that is 3-transitive. 


It is better to have an option for randomization. We know that choosing 2 elements at random in a simple Lie group leads to expander family of Cayley graphs. 


Is the same thing true for example, in case we randomize elements in Schreier graph of $\mathrm{PSL}\_2$ acting on the projective plane? 


Let's ask this formally: Given $\epsilon$, is there a family of bounded degree Schreier graphs with a 3-transitive group action such that if I randomize x generators, I have $\epsilon$-expansion in probability p independent of n(the number of vertices of the graph). 


I am looking for a known result similar to this. Or maybe references. 


And if not 3, then 2-transitive would be OK.


Thanks



Examples of 3-transitive expander family of Schreier graphs
Well, it was solved.


It is a bit long for here but given $G=SL(2,p)$, $Cay(G, S) $ expander, 
we can see that $Sc:=Sch[G,P^1(F\_p),S]$ is also an expander, by comparing their mixing time and find it is less than that of the cayley graph. 


The action is 3-transitive because the action of $GL$ is. 


The idea is that $s\_t ... \cdot s\_1 $ is mixed after $t$ step and that is what acts on any element in the Scherier graph. We divide it into classes in which $s\_t ... s\_1 \cdot x = y$ , and use the mixing condition. 


