## Notes of Arindam Biswas’ informal Cambridge lecture 28-02-2017

Spectral gaps for ${Sl(2,F_p)}$

1. Context

Goal: produce expanders. These will be Cayley graphs of finite quotients of a group, ${Sl(2,{\mathbb Z})}$, which does not have property (T). Therefore, the method is new. Lubotsky had pointed out that, prior to Bourgain-Gamburd’s work, one did not know wether the following generating systems

$\displaystyle \begin{array}{rcl} S^i_p =\{\begin{pmatrix} 1 & i \\ 0 & 1 \end{pmatrix},\begin{pmatrix} 1 & 0 \\ i & 1 \end{pmatrix}\} \end{array}$

produce an expander, as ${p\rightarrow\infty}$, when ${i=3}$. When ${i=1}$ or 2, it follows from Selberg’s ${\frac{3}{16}}$ theorem.

Theorem 1 (Bourgain-Gamburd) Fix a finite subset ${S}$ of ${SL(2,{\mathbb Z})}$. Let ${S_p}$ be the image of ${S}$ in ${G_p:=Sl(2,F_p)}$. Then the family ${Cay(G_p,S_p)}$ is an expander iff ${S}$ generates a non-elementary subgroup of ${Sl(2,{\mathbb Z})}$.

${\Rightarrow}$ is straightforward.

2. Scheme of proof

The main result is

Proposition 2 Fix ${k\geq 2}$. Suppose that none of the ${k}$ generators is an involutions and that

$\displaystyle \begin{array}{rcl} \mathrm{girth}(Cay(G_p,S_p))\geq|tau\,\log_{2k}p. \end{array}$

Then ${Cay(G_p,S_p)}$ form an expander.

Indeed, Gamburd had observed earlier that the girth is at least logarithmic, when ${S}$ is free. It follows from a short argument going back to Margulis.

3. Connectedness of ${Cay(G_p,S_p)}$

This follows from Dickson’s classification of subgroups of ${G_p}$. If ${p\geq 5}$, proper subgroups either

1. contain a cyclic subgroup of index 2,
2. have order ${\leq 120}$,
3. or lie in a Borel subgroup.

All such subgroups (but those of order ${\leq 120}$) satisfy the following law: all brackets ${[[g_1,g_2],[g_3,g_4]]}$ vanish. This implies that the girth of a non-generating subset ${S_p}$ would stay bounded, contradiction.

4. Measures

We consider probability measures on ${G_p}$ and study how their ${\ell^2}$ norm decays under convolution. We then apply this to the uniform measure ${\mu_S}$ on ${S_p}$. It yields information on random walks, hence on spectral gap.

Indeed, let ${W_{2\ell}}$ denote the number of walks on ${Cay(G_p,S_p)}$. Let ${T}$ be the adjacency matrix of this graph and ${\lambda_0,\ldots,\lambda_{N-1}}$ its eigenvalues. Its size is ${N=|G_p|=\frac{1}{2}(p^2-1)}$. Then

1. ${NW_{2\ell}=\mathrm{trace}(T^{2\ell})=\sum_{i}\lambda_i^{2\ell}}$.
2. ${\mu_S^{2\ell}(1)=\frac{W_{2\ell}}{(2k)^{2\ell}}}$.
3. ${\mu_S^{2\ell}}$ is symmetric.

The main point in the proof is the following Lemma.

Lemma 3 Let ${|S_p|=2k}$. Assume that ${Cay(G_p,S_p)}$ has logarithmic girth. Then for all ${\epsilon>0}$, there exists ${C(\epsilon,\tau)}$ such that for ${\ell\geq C}$,

$\displaystyle \begin{array}{rcl} \|\mu_S^{\ell}\|_2 \leq p^{-\frac{3}{2}+\epsilon}. \end{array}$

4.1. Here is how the Lemma implies the Proposition

The Lemma implies that, for ${\ell\geq C\,\log_{2k}p}$,

$\displaystyle \begin{array}{rcl} W_{2\ell}\leq (2k)^{2\ell}p^{-3+2\epsilon}. \end{array}$

We know that the every nontrivial representation of ${G_p}$ occurs in ${\ell^2(G_p)}$ with multiplicity at least ${\frac{1}{2}(p-1)}$ (Frobenius). The trivial representation occurs only for ${\lambda_0}$. Therefore ${\lambda_1}$ has multiplicity at least ${\frac{1}{2}(p-1)}$. Hence

$\displaystyle \begin{array}{rcl} \lambda_1^{2\ell}\leq N\frac{(2k)^{2\ell}}{\frac{1}{2}(p-1)p^{3-2\epsilon}}\leq const.\frac{(2k)^{2\ell}}{p^{-2\epsilon}} \end{array}$

Choosing ${2\ell= C\,\log_{2k}p}$ yields a constant lower bound on ${\lambda_0-\lambda_1}$, whence the expansion.

5. Decay of ${\ell^2}$ norms

Lemma 4 Fix ${\gamma<\frac{3}{4}}$. Assume a probability measure satisfies

1. ${\|\nu\|_\infty,
2. ${\|\nu\|_2\geq p^{-\frac{3}{2}+\gamma}}$,
3. for every proper subgroup ${G_0}$,

$\displaystyle \begin{array}{rcl} \nu\star\nu(G_0)\leq p^{-\gamma}. \end{array}$

Then

$\displaystyle \begin{array}{rcl} \|\nu\star\nu\|_2 \leq p^{-\epsilon}\|\nu\|_2. \end{array}$

The proof of this relies on Helfgott’s product theorem and approximate subgroups.

First, one splits ${G_p}$ into level sets of ${\nu}$ and approximates ${\nu}$ with ${\tilde\nu}$ which takes values in powers of ${1/2}$. Then, using triangle inequality, the estimate boils down to estimating the convolution of two characteristic functions of subsets.

Define multiplicative energy

$\displaystyle \begin{array}{rcl} E(A,B)=|\{(x_1,x_2,y_1,y_2)\in A^2\times B^2\, x_1y_1=x_2y_2\}|=\|\chi_A \star\chi_B\|_2^2. \end{array}$

If Lemma fails, we find subsets ${A}$ and ${B}$ such that

$\displaystyle \begin{array}{rcl} E(A,B)\geq p^{-4\epsilon}|A|^{3/2}|B|^{3/2}. \end{array}$

Then, Balog-Szemeredi-Gowers’ theorem implies that ${A}$ contains a subset ${X}$ such that ${|X|\geq c|A|}$ and ${|XX^{-1}|\leq C\,|A|}$. Thus there is an approximate subgroup ${H}$ of comparable size, i.e. at most ${p^{3-2\gamma}}$.

Helfgott’s product theorem states that if ${|H|\leq p^{3-\gamma}}$ and ${H}$ is not contained in a proper subgroup, then ${|HHH|\geq c|H|^{1+\kappa}}$. This gives the required contradiction, and proves Lemma 4.

6. Proof of Lemma 3

The girth assumption implies that walks of length ${\tau\log_{2k}p}$ behave like in a ${2k}$-regular tree, where ${\ell^2}$ norms decay exponentially under convolution (Kesten).

Lemma 4 is used iteratively to pass from time scale ${\tau\log_{2k}p}$ to ${C\,\log_{2k}p}$.

7. Final remarks

In a finite group ${G}$, assume that one has the following bounds:

1. Lower bound on degrees of nontrivial representations: ${\geq |G|^\beta}$.
2. Classification of approximate subgroups: ${|G|^\delta}$-approximate subgroup either have size ${\geq |G|^{1-\epsilon}}$ or do not differ much from cosets of subgroups.
3. Non-concentration estimate for convolutes on subgroups: ${\mu_S^n(H)\leq [G:H]^{-\kappa}}$.

Then ${\lambda_1(Cay(G,S))\geq \beta e^{-C/\delta}}$.

Thus fact 3 (non-concentration) is the suitable generalization of the girth bound.

Benoist-de Saxce implement a similar strategy in the continuous setting. Their almost Diophantine assumption is a non-concentration estimate.

They conjecture that every probability measure generating a dense subgroup is almost Diophantine. They prove this for measures whose support is contained in matrices with algebraic entries (this relies on Benoist-Quint).