Removing the polynomial prefactor from the finite-time lattice count in strata (Theorem 5.1 of Eskin–Mirzakhani–Rafi)

In 'Counting closed geodesics in strata' (Inventiones mathematicae, 2019), Eskin, Mirzakhani, and Rafi prove their asymptotic count of closed Teichmüller geodesics in a stratum of quadratic differentials through a finite-time lattice count, their Theorem 5.1: the number of mapping-class-group orbit points of $Y$ reachable from $X$ by a Teichmüller geodesic segment of length at most $\tau$ contained in the stratum, with endpoint triangulations sharing $j$ homologically independent saddle connections, is bounded (up to a uniform multiplicative constant) by $\tau^{|S_X|+|S_Y|}e^{(h-j)\tau}G(X)G(Y)$, where $S_X$ is the set of short curves on $X$ and $G(X)$ is the corresponding product of inverse square roots of their extremal lengths. In Remark 5.2(2) the authors state that the result 'appears to be correct even without the term $\tau^{|S_X|+|S_Y|}$' but that the proof 'would become significantly more complicated', and they leave the sharper bound unproved. The source statement is an upper bound only, not a two-sided asymptotic. The question of whether the polynomial prefactor can be removed from the upper bound of Theorem 5.1 remains open.

Background

Let $S=S_{g,p}$ be an oriented surface of genus $g$ with $p$ punctures, with $3g-3+p>0$. The Teichmüller space $T(S)$ is the space of marked conformal structures on $S$; the mapping class group $\Gamma(S)$, the group of isotopy classes of orientation-preserving homeomorphisms of $S$, acts on $T(S)$ by changing the marking, and the quotient $M(S)=T(S)/\Gamma(S)$ is the moduli space of Riemann surfaces. The cotangent bundle of $T(S)$ is identified with the space of quadratic differentials: a quadratic differential $q$ is a holomorphic section of the square of the cotangent bundle, and it endows the underlying Riemann surface $X$ with a flat metric with conical singularities at the zeros of $q$. The Teichmüller geodesic flow is the diagonal flow $g_t$ on the bundle $Q^1T(S)$ of unit-area quadratic differentials; its orbits project to geodesics of the Teichmüller metric on $T(S)$. Fixing a singularity type $\sigma$ (the orders of zeros and poles), the stratum $\mathcal{Q}(\sigma)\subset Q^1T(S)$ is the locally closed submanifold of unit-area quadratic differentials with singularity pattern exactly $\sigma$; the flow preserves it.

A foundational counting result of Eskin and Mirzakhani states that the number $N(R)$ of closed Teichmüller geodesics of length at most $R$ in the full moduli space grows like $e^{hR}/(hR)$ with $h=6g-6+2p$, the analogue for the Teichmüller flow of Margulis' theorem for closed geodesics on negatively curved manifolds (this built, in turn, on Mirzakhani's thesis count of simple closed geodesics on hyperbolic surfaces). Eskin, Mirzakhani, and Rafi extended such counting to each stratum: for a connected component $\mathcal{C}$ of the projection of $\mathcal{Q}(\sigma)$ to moduli space, the number $N(\mathcal{C},R)$ of closed geodesics of length at most $R$ in $\mathcal{C}$ is asymptotic to $\frac{e^{hR}}{hR}$ with $h=(1+\dim_{\mathbb{R}}\mathcal{C})/2$. The essential difficulty is that the Teichmüller flow is not uniformly hyperbolic: it degenerates near loci where saddle connections (flat geodesic segments joining singularities with no singularity in their interior) become short. Their proof combines a lattice point count restricted to the stratum with an exponential-recurrence statement showing that geodesics spending a definite fraction of their time near the degenerating locus are exponentially rare.

The question concerns the finite-time combinatorial input to that argument, Theorem 5.1 of the paper. For a point $X\in T(S)$, let $S_X$ denote the set of $\varepsilon_0$-short curves on $X$: simple closed curves $\alpha$ whose extremal length $\mathrm{Ext}_X(\alpha)$ — a conformal invariant of the curve on the Riemann surface — is at most a fixed small uniform constant $\varepsilon_0$. Curves that are short on $X$ are pairwise disjoint, and $S_X$ is empty exactly when $X$ lies in a fixed compact part of moduli space. Define $G:T(S)\to\mathbb{R}^+$ by $G(X)=1+\prod_{\alpha\in S_X}\mathrm{Ext}_X(\alpha)^{-1/2}$ (equal to $2$ when $S_X$ is empty), a proper function that blows up as $X$ approaches the cusp of moduli space along short-curve pinches. For $X,Y\in T(S)$ and $\tau>0$, let $B(\mathcal{Q}(\sigma),X,\tau)$ be the set of points $Z\in T(S)$ such that there is a Teichmüller geodesic segment of length at most $\tau$, contained in the stratum $\mathcal{Q}(\sigma)$, starting and ending within a fixed distance $r_0$ of $X$ and of $Z$. One should think of $B(\mathcal{Q}(\sigma),X,\tau)$ as a ball of radius $\tau$ about $X$ in which movement is constrained to directions tangent to the stratum. Let $B_j(\mathcal{Q}(\sigma),X,\tau)\subset B(\mathcal{Q}(\sigma),X,\tau)$ be the subset where the initial and terminal quadratic differentials admit $(q,\tau)$-regular triangulations — triangulations of the complement of the large flat cylinders of $q$ by disjoint saddle connections, with the flat length of each edge bounded, up to a uniform multiplicative constant, by the size of each thick subsurface it crosses, and with uniformly bounded twisting around short cylinder curves (Definition 3.11 of the source) — sharing at least $j$ homologically independent saddle connections (a saddle connection determines a class in the relative homology of the surface modulo the singularities of $q$, and a family of saddle connections is homologically independent when those classes are linearly independent), and set $B_j(\mathcal{Q}(\sigma),X,Y,\tau)=B_j(\mathcal{Q}(\sigma),X,\tau)\cap\Gamma(S)\cdot Y$. Writing $h=\frac{1}{2}(1+\dim_{\mathbb{R}}\mathcal{Q}(\sigma))$ — the same exponent as $h=(1+\dim_{\mathbb{R}}\mathcal{C})/2$ in the asymptotic count above, since each connected component $\mathcal{C}$ of the projection of $\mathcal{Q}(\sigma)$ to moduli space has the same real dimension as $\mathcal{Q}(\sigma)$ — Theorem 5.1 states

$$B_j(\mathcal{Q}(\sigma),X,Y,\tau)\stackrel{*}{\prec}\tau^{|S_X|+|S_Y|}\,e^{(h-j)\tau}\,G(X)G(Y),$$

where $A\stackrel{*}{\prec}B$ means $A\le cB$ for a constant $c$ depending only on the topology of $S$ and the fixed constants $\varepsilon_0,r_0$, uniform in $X$, $Y$, $\tau$, and $j$. In the proof, each point $Z$ of $B_j$ is mapped to a marking (a partial triangulation together with short-curve length data and twist parameters) of bounded length and $\tau$-bounded twist in $X$; the prefactor $\tau^{|S_X|}$ arises from Lemma 5.6, which bounds the number of such markings by $\tau^{|S_X|}$ because the marking may twist up to on the order of $\tau$ around each short curve, and the factor $\tau^{|S_Y|}$ arises from the analogous multiplicity control at the $Y$ end. In Remark 5.2(2) the authors note: 'The statement appears to be correct even without the term $\tau^{|S_X|+|S_Y|}$. However, the proof would become significantly more complicated.' In the paper's own applications the polynomial prefactor is harmless (the time horizon $\tau$ is fixed and only the segment length $R\to\infty$ matters, so the prefactor is absorbed into constants), so the sharpened statement was left as an explicitly identified but unproved improvement.

Problem Statement

Determine whether the polynomial prefactor in Theorem 5.1 of Eskin–Mirzakhani–Rafi can be removed; that is, prove or disprove the following statement. Let $S=S_{g,p}$ be an oriented surface of genus $g$ with $p$ punctures, let $\mathcal{Q}(\sigma)\subset Q^1T(S)$ be a stratum of unit-area quadratic differentials of singularity type $\sigma$ on Teichmüller space, let $h=\frac{1}{2}(1+\dim_{\mathbb{R}}\mathcal{Q}(\sigma))$, and let $\Gamma(S)$ be the mapping class group. For $X,Y\in T(S)$ and $\tau>0$, let $B(\mathcal{Q}(\sigma),X,\tau)$ be the set of points $Z\in T(S)$ for which there exists a Teichmüller geodesic segment of length at most $\tau$, contained in $\mathcal{Q}(\sigma)$, starting within a fixed distance $r_0$ of $X$ and ending within $r_0$ of $Z$; let $B_j(\mathcal{Q}(\sigma),X,\tau)$ be the subset for which the endpoint quadratic differentials admit $(q,\tau)$-regular triangulations sharing at least $j$ homologically independent saddle connections; and let $B_j(\mathcal{Q}(\sigma),X,Y,\tau)=B_j(\mathcal{Q}(\sigma),X,\tau)\cap\Gamma(S)\cdot Y$. As in the source, write $S_X$ for the set of $\varepsilon_0$-short curves on $X$ and $G(X)=1+\prod_{\alpha\in S_X}\mathrm{Ext}_X(\alpha)^{-1/2}$. The question is whether the sharpened upper bound

$$|B_j(\mathcal{Q}(\sigma),X,Y,\tau)|\stackrel{*}{\prec}e^{(h-j)\tau}\,G(X)G(Y)$$

holds, where $\stackrel{*}{\prec}$ means up to a multiplicative constant depending only on the topology of $S$ and the fixed constants $\varepsilon_0,r_0$ (uniform in $X,Y\in T(S)$, in $\tau$, and in the admissible values of $j$), with no additional factor polynomial in $\tau$. The sharpened bound holds precisely when the ratio $|B_j(\mathcal{Q}(\sigma),X,Y,\tau)|/(e^{(h-j)\tau}G(X)G(Y))$ remains bounded, and fails precisely when that ratio is unbounded along some sequence of parameters (a stratum, endpoints, and time horizons $\tau\to\infty$). The source theorem is an upper bound only.

References

1. Alex Eskin, Maryam Mirzakhani, and Kasra Rafi, "Counting closed geodesics in strata", Inventiones mathematicae 215 (2019), no. 2, 535–607. DOI: 10.1007/s00222-018-0832-y, https://doi.org/10.1007/s00222-018-0832-y (arXiv:1206.5574, https://arxiv.org/abs/1206.5574, "final version before publication")
2. John Rached, "Counting Closed Geodesics in Rank 1 SL(2,R)-orbit Closures", preprint, 2022. arXiv:2207.00073, https://arxiv.org/abs/2207.00073
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