# Examples of 3-transitive expander family of Schreier graphs

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

• There are known examples of such families (e.g. using Brooks' spectral gap for congruence actions of $\mathrm{SL}_2(\mathbf{Z})$). Could you be more precise? to me the question is presently written in a too informal style to really know what you wish to have. – YCor Apr 17 at 19:25
• Added formality. I don't know what results are there, so I have asked that informally. Anything close to what I wrote would be nice. If you need to take specific generators for each $n$ , that would be interesting too. – user2679290 Apr 17 at 19:54
• So you're looking for a family $(G_i,X_i)$ with $G_i$ a finite group acting on $X_i$ (3-transitively) such that for some $x$ and each $\varepsilon>0$, the probability $p_i=p_i(x,\varepsilon)$ that a random $x$-tuple in $G_i$ (using uniform probability on $G_i^x$) is generates $G_i$ with $\varepsilon$ spectral gap on $\ell^2(X_i)$, satisfies $\liminf p_i>0$. (This is not exactly what you're asking, but roughly is it?) – YCor Apr 17 at 20:07
• With predetermined generators, I mentioned Brooks' result: hence for each fixed generating subset of $\mathrm{SL}_2(\mathbf{Z})$, you have expansion for the Schreier graphs on $\mathbb{P}^1(\mathbf{Z}/p\mathbf{Z})$ when $p$ ranges over primes. – YCor Apr 18 at 22:06
• I thinks it's: R. Brooks, The ﬁrst eigenvalue in a tower of coverings. Bull. Amer. Math. Soc. 13 (1985),no. 2, 137-140. But actually the spectral gap itself for congruence subgroups is rather due to Selberg: On the estimation of Fourier coefficients of modular forms, Proc. Symp. Pure Math. VII, Amer. Math. Soc.(1965), 1-15. – YCor Apr 18 at 22:48

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.
• Actually for a field $K$ the $\mathrm{SL}_2(K)$-action on distinct triples of $\mathbb{P}^1(K)$ has its orbits indexed by ${K^*}/{K^*}^2$, so is not transitive in general (for a finite field of odd characteristic it has 2 orbits). That is, it's not 3-transitive, albeit quite close. If $-1$ is not a square in $K$ (i.e., if $K$ has cardinal in $4\mathbf{Z}+3$), the action is however transitive on 3-element subsets, and also in this case the $\mathrm{GL}_2(\mathbf{Z})$-action is 3-transitive. – YCor Apr 20 at 0:56