In this previous post here (from November 2009), I mentioned some discussions I had with Artur Avila and Jean-Christophe Yoccoz about the spectral properties of the natural -representation associated to Teichmüller curves. In particular, as an outcome of this discussion, we observed that the existence of (arithmetic) Teichmüller curves whose natural -representation exhibits complementary series follows from the combination of:

- Ratner’s estimate of the rate of mixing of the geodesic flow on hyperbolic surfaces in terms of the spectral properties (namely, the size of the “spectral gap”) of appropriate -representations,

- a certain cyclic cover construction (going back to A. Selberg, as it was pointed out to us by P. Hubert and N. Bergeron) to slow down the rate of mixing of the geodesic flow and,
*a fortiori*, to decrease the size of the spectral gap (by Ratner’s estimate above),

- a recent theorem of J. Ellenberg and D. B. McReynolds on arithmetic Veech sublattices of implying that the hyperbolic surfaces obtained by the cyclic cover construction above can be realized as (arithmetic) Teichmüller curves (i.e., projections in the moduli space of curves of -orbits of square-tiled surfaces).

Of course, a natural question coming from this existence result is: can one construct an *explicit* square-tiled surface (origami) whose Teichmüller curve has complementary series? Evidently, the naive approach would be to try to make explicit all steps of the previous arguments. However, there are at least two sources of difficulties when one try to do so:

- As one can see by the end of this post here, Ratner’s estimate applied to the hyperbolic surfaces obtained by the cyclic covering of degree mentioned above says that the absence of complement series forces , where, in principle, the implied constant here can be computed explicitly (we’ll come back to this point later). In other words, one sees complementary series as soon as is large. But, since grows slowly (and the implied constant is not particularly small), one can’t hope to get complementary series for free.

- After getting the hyperbolic surface with complementary series, one still has to realized it as a Teichmüller curve, that is, one has to enter into the arguments of J. Ellenberg and D. McReynolds. In particular, this means that we have to consider more covering constructions, so that the number of squares of the final origami (which is certainly related to the degrees of these covering constructions) is likely to be large.

Fortunately, it turns out that one can modify this strategy to get explicit origamis with complementary series. Indeed, in this note here (uploaded to ArXiv a few days ago), Gabriela Schmithüsen and I observed that:

- the estimate coming from Ratner’s estimate (in the absence of complementary series) can be “improved” to in the case of the cyclic cover construction at hand because of the so-called Buser’s inequality (relating the first eigenvalue of the Laplacian on the hyperbolic surface to the Cheeger constant).
- since the first eigenvalue doesn’t increase by taking coverings, we don’t have to employ Ellenberg-McReynolds’ method
*jusqu’au but*: it suffices to construct an origami whose Teichmüller curve is a*covering*of the hyperbolic surface provided by the previous item, that is, one only needs an origami whose Veech group (fundamental group of its Teichmüller curve) is a finite-index*subgroup*of a certain*target group*(namely, the fundamental group of the hyperbolic surface). From the technical point of view, this simplifies the discussion since a lot of the effort in Ellenberg-McReynolds article concerns the realization of*every*finite-index subgroup of the principal congruence of level two containing as the fundamental group of a Teichmüller curve (obviously a much harder task than realizing some finite-index subgroup of a certain fixed target group). In particular, in our note, we simply combine some parts of Ellenberg-McReynolds methods with some arguments in Gabriela’s thesis.

As a by-product of these arguments, we check that our “smallest” example of arithmetic Teichmüller curve with complementary series is associated to an origami with 576 squares and genus 147 (indeed, explicit presentations of this origami appear in Corollary 4.3 and Appendix B of our note).

As the reader can verify, the note by Gabriela and myself contains plenty of details and illustrations around the arguments mentioned above (and that’s why it has 40 pages despite our initial [frustrated] intention of keeping it within the usual standards for a “note” [whatever this means]).

Closing this post, let me say that I’ll not try to give more explanations about this note with Gabriela here. Instead, I will transcript in a future post some personal notes about explicit constants in Ratner’s estimate of rate of mixing of geodesic flows on hyperbolic surfaces. Indeed, even though these notes were not useful in this particular project (as the knowledge of explicit constants is not sufficient to beat the decay), I think that they may help others. Also, since the explicit calculation of constants in Ratner’s result is straightforward (and very simple minded), they’re not publication-quality, so that this blog is the ideal place for them.

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