Spectral gaps for
Goal: produce expanders. These will be Cayley graphs of finite quotients of a group, , 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
produce an expander, as , when . When or 2, it follows from Selberg’s theorem.
Theorem 1 (Bourgain-Gamburd) Fix a finite subset of . Let be the image of in . Then the family is an expander iff generates a non-elementary subgroup of .
2. Scheme of proof
The main result is
Proposition 2 Fix . Suppose that none of the generators is an involutions and that
Then form an expander.
Indeed, Gamburd had observed earlier that the girth is at least logarithmic, when is free. It follows from a short argument going back to Margulis.
3. Connectedness of
This follows from Dickson’s classification of subgroups of . If , proper subgroups either
- contain a cyclic subgroup of index 2,
- have order ,
- or lie in a Borel subgroup.
All such subgroups (but those of order ) satisfy the following law: all brackets vanish. This implies that the girth of a non-generating subset would stay bounded, contradiction.
We consider probability measures on and study how their norm decays under convolution. We then apply this to the uniform measure on . It yields information on random walks, hence on spectral gap.
Indeed, let denote the number of walks on . Let be the adjacency matrix of this graph and its eigenvalues. Its size is . Then
- is symmetric.
The main point in the proof is the following Lemma.
Lemma 3 Let . Assume that has logarithmic girth. Then for all , there exists such that for ,
4.1. Here is how the Lemma implies the Proposition
The Lemma implies that, for ,
We know that the every nontrivial representation of occurs in with multiplicity at least (Frobenius). The trivial representation occurs only for . Therefore has multiplicity at least . Hence
Choosing yields a constant lower bound on , whence the expansion.
5. Decay of norms
Lemma 4 Fix . Assume a probability measure satisfies
- for every proper subgroup ,
The proof of this relies on Helfgott’s product theorem and approximate subgroups.
First, one splits into level sets of and approximates with which takes values in powers of . Then, using triangle inequality, the estimate boils down to estimating the convolution of two characteristic functions of subsets.
Define multiplicative energy
If Lemma fails, we find subsets and such that
Then, Balog-Szemeredi-Gowers’ theorem implies that contains a subset such that and . Thus there is an approximate subgroup of comparable size, i.e. at most .
Helfgott’s product theorem states that if and is not contained in a proper subgroup, then . This gives the required contradiction, and proves Lemma 4.
6. Proof of Lemma 3
The girth assumption implies that walks of length behave like in a -regular tree, where norms decay exponentially under convolution (Kesten).
Lemma 4 is used iteratively to pass from time scale to .
7. Final remarks
In a finite group , assume that one has the following bounds:
- Lower bound on degrees of nontrivial representations: .
- Classification of approximate subgroups: -approximate subgroup either have size or do not differ much from cosets of subgroups.
- Non-concentration estimate for convolutes on subgroups: .
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).