## The French Andes and spectral gaps

I have spent this last week in a maths conference at Peyresq, in the French alps. I am tempted to say “the French Andes”: an altitude of 1500m is not trivial at this latitude, and the climb from the sea (two hours by bus) reminded me of the highway from the Pacific to Ayacucho. The village is up on the hill, on a rocky outcrop.

Peyresq was once a nearly completely depopulated village. (It still is, in the sense that it seems to have no more than ten permanent inhabitants.) It was rebuilt in the 1950s/1960s by a group of Belgian academics that redefined it as, in essence, a conference centre.

It is impossible not to think that there are many depopulated villages in the Peruvian Andes, and that one of them could be rebuilt to gather mathematicians from South America and elsewhere. There are two tentative objections one could raise to this. While Peyresq was built by individuals,
the State helped with the project by providing basic infrastructure: running water, electricity, telephone lines. A similar proyect could succeed elsewhere only with similar state support. (Of course, this would be splendid if there were still a bit of a local population that would get basic services it has a right to as a result.)

The other issue consists of the reason why many villages in the Andes were depopulated. One thing to avoid is, say, Lidice Mathfest. This is a complicated matter; Peyresq itself disappeared as a living settlement arguably because the death of young people in two world wars took it below a critical mass (see the legend in the picture above).

One thing is certain, though: when a mathematics meeting is held at a place that is tempting for hiking, the local authorities better be good at finding lost mathematicians. (No, it did not happen to me…)

(Oh, another request: peoples of the world, do not follow the Chilean lead in building unheated places with large windows in the middle of mountain ranges (so as to enhance the moral fibre of the guests?).)

Une petite question aux lecteurs spectraux. In any graph, the existence of a spectral gap implies small diameter. In a Cayley graph of bounded degree, the implication goes in the opposite direction as well: given a group $G$ and a set of generators $A$, we have
$|\lambda_0 - \lambda_1|\geq \frac{1}{|A| diam(\Gamma(G,A))^2},$ where $\lambda_0\leq \lambda_1\leq \dotsb$ are the eigenvalues of the discrete Laplacian $\Delta$ on the Cayley graph $\Gamma(G,A)$.

Question: does a similar result hold for Schreier graphs? It would seem to me that the proof in Diaconis and Saloff-Coste, Comparison techniques for random walks on groups, Ann. of Prob., v. 21, no. 4 (1993), p. 2138 (not the first proof; see the references there) carries over to Schreier graphs without much trouble. I am a little suspicious, however, since I would expected such a general result to be already in the literature if true and easily provable. (Either the premise or the reasoning in this meta-mathematical reasoning are faulty, or I have not looked hard enough…)

(If this does hold, then Schreier graphs are esperantist (a nice recent coinage) if only if they have polylogarithmic diameter. This is already known for Cayley graphs, by what I said above.)

I am a number theorist with side interests in combinatorics and group theory.
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### 6 Responses to The French Andes and spectral gaps

1. Ricardo Menares says:

Part of this post can be easily generalized: don’t follow the Chilean lead in anything at all…

• valuevar says:

Hah – may I count this as part of the Chilean national character showing through?

• Ricardo Menares says:

As you have surely noticed by this point, induction on one case is often misleading (like Chileans!)

• valuevar says:

Not to mention the fact that they do not seem to have a natural ordering.

• (Apparent, quite convincing) Lack of natural order is just a minor technical problem. Since Chileans form a finite set, just pick some order (without axiom of choice). The fundamental problem is the induction step.

2. valuevar says:

To answer my own question: small diameter implies small mixing time only for connected vertex-transitive graphs. This includes all Cayley graphs but not most Schreier graphs. (Schreier graphs can be pretty much anything.) Thanks to Laszlo Babai (one of the people to whom this result is due!) and Igor Pak for pointing this out.