Sharp error terms and uniformity in the centers in the Teichmuller lattice point and volume asymptotics

Athreya, Bufetov, Eskin, and Mirzakhani (Duke Math. J. 2012) proved that the number of mapping class group orbit points of Teichmuller space inside a Teichmuller metric ball of radius $R$ grows like a constant times $e^{hR}$ with $h=6g-6$, together with sector, bisector, and ball-volume analogues, the leading constants involving the Hubbard-Masur function $\Lambda$, and left several quantitative limitations open. Later literature has resolved two of them: $\Lambda$ is in fact a genus-only constant $\Lambda_g$ (Mirzakhani's argument, written out by Dumas, Acta Math. 2015), and qualitative power-saving error terms exist for centers restricted to a compact set (Arana-Herrera, Duke Math. J. 2023). The open problem is the completion of this quantitative theory: determine the optimal error terms and any secondary asymptotic expansion for the lattice, sector, bisector, and volume counts; make the effective estimates uniform in the pair of centers as they degenerate toward the boundary of Teichmuller space, with sharp control of the recurrence data in the polar-coordinates estimate; and quantify sharply the subleading contributions of neighborhoods of the multiple zero locus.

Background

Fix $g\ge 2$ and a closed oriented surface $\Sigma_g$ of genus $g$. The Teichmuller space $\mathcal{T}_g$ is the space of marked conformal structures on $\Sigma_g$ up to isotopy; the mapping class group $\mathrm{Mod}_g$, the group of isotopy classes of orientation-preserving diffeomorphisms of $\Sigma_g$, acts on $\mathcal{T}_g$ by changing the marking, and the quotient $\mathcal{M}_g$ is the moduli space of genus-$g$ Riemann surfaces. The Teichmuller metric $d_{\mathcal{T}}$ on $\mathcal{T}_g$ is the Finsler metric measuring optimal quasiconformal distortion; Kerckhoff's formula expresses $d_{\mathcal{T}}(X,Y)$ as a supremum of logarithms of ratios of extremal lengths. The cotangent space at $X\in\mathcal{T}_g$ identifies with the space $Q(X)$ of holomorphic quadratic differentials, tensors locally of the form $\varphi(z)\,dz^2$; $Q^1\mathcal{T}_g$ denotes the bundle of unit-area quadratic differentials. A quadratic differential $q$ induces a singular flat metric $|q|^{1/2}$ whose zeros are cone-type singularities; the principal stratum $Q^1\mathcal{T}_g(1)$ consists of those $q$ with all zeros simple, and its complement is the multiple zero locus. The Masur-Veech measure $\mu$ is the natural smooth measure on $Q^1\mathcal{T}_g$, finite on the quotient $Q^1\mathcal{M}_g$; write $m=\pi_*\mu$ for its pushforward to $\mathcal{T}_g$ under the projection $\pi:Q^1\mathcal{T}_g\to\mathcal{T}_g$.

Teichmuller geodesics are projections of orbits of the Teichmuller geodesic flow $g_t$, the action of the diagonal subgroup of $SL_2(\mathbb{R})$ on quadratic differentials. In the normalization of the source paper, $d_{\mathcal{T}}(\pi(q),\pi(g_tq))=t$ and the entropy of the flow with respect to $\mu$ is $h=6g-6$; the flow is exponentially mixing on the principal stratum (correlations between sufficiently regular observables decay exponentially in the time separation). A measured foliation is a singular foliation together with a transverse measure, considered up to Whitehead equivalence (the equivalence generated by isotopy together with splitting or merging of singular leaves); real and imaginary measured foliations $\eta^\pm(q)$ record the horizontal and vertical foliations of $q$, and the space $\mathcal{ML}_g$ of measured geodesic laminations (identified with the space $\mathcal{MF}_g$ of measured foliations) is a piecewise-linear manifold of dimension $6g-6$ whose integer points are multicurves (finite nonnegative integral combinations of pairwise disjoint simple closed curves), carrying the Thurston measure $\nu$, the Lebesgue-class measure induced by this integral structure. The extremal length $\mathrm{Ext}_\beta(X)$ of $\beta\in\mathcal{ML}_g$ at $X$ is the conformal invariant $\sup_\rho \ell_\beta(\rho)^2/\mathrm{Area}(X,\rho)$ over metrics $\rho$ conformal to $X$; the Hubbard-Masur theorem states that for each $X$ and $\beta$ there is a unique holomorphic quadratic differential on $X$ with horizontal measured foliation $\beta$, and its flat area equals $\mathrm{Ext}_\beta(X)$.

In his thesis, Margulis proved that for compact negatively curved manifolds the orbit count of a deck group in a ball of radius $R$ of the universal cover is asymptotic to $c\,e^{hR}$, with angular refinements governed by the Bowen-Margulis conformal densities (the flow-equivariant measures on the sphere at infinity describing the distribution of endpoints of geodesic rays). Athreya, Bufetov, Eskin, and Mirzakhani adapted this program to Teichmuller space, with $\mathrm{Mod}_g$ in place of the fundamental group and the pair of projective foliation measures in place of the sphere at infinity. Their main results are: the lattice point asymptotic $|\mathrm{Mod}_g\cdot Y\cap B_R(X)|\sim\frac{\Lambda(X)\Lambda(Y)}{h\,m(\mathcal{M}_g)}\,e^{hR}$; the volume asymptotic $m(B_R(X))\sim\frac{\Lambda(X)\int_{\mathcal{M}_g}\Lambda\,dm}{h\,m(\mathcal{M}_g)}\,e^{hR}$; and sector and bisector refinements, defined as follows. The boundary at infinity of Teichmuller space identifies with the space $PMF_g$ of projective measured foliations, and for $X\in\mathcal{T}_g$ an open set $U\subset PMF_g$ corresponds to a set $U(X)$ of unit directions at $X$, under the identification of the unit sphere $S(X)=\pi^{-1}(X)\subset Q^1\mathcal{T}_g$ with projective measured foliations given by the Hubbard-Masur theorem; the sector based at $X$ in the direction $U$ is $\mathrm{Sect}_U(X)=\bigcup_{t\ge0}\pi(g_tU(X))\subset\mathcal{T}_g$. Writing $s_X$ for the conditional measure on $S(X)$ obtained by disintegrating $\mu$ along the fibers of $\pi$, the sector count satisfies $|\mathrm{Sect}_U(X)\cap B_R(X)\cap\mathrm{Mod}_g\cdot Y|\sim\frac{\Lambda(Y)}{h\,m(\mathcal{M}_g)}\,e^{hR}\int_{U(X)}\lambda^-(q)\,ds_X(q)$ and the bisector count satisfies $\#\{\gamma\in\mathrm{Mod}_g:\gamma\cdot Y\in B_R(X)\cap\mathrm{Sect}_U(X),\ \gamma^{-1}\cdot X\in\mathrm{Sect}_V(Y)\}\sim\frac{e^{hR}}{h\,m(\mathcal{M}_g)}\left(\int_{U(X)}\lambda^-\,ds_X\right)\left(\int_{V(Y)}\lambda^+\,ds_Y\right)$, where $\lambda^\pm$ are the Hubbard-Masur functions, the Radon-Nikodym derivatives between $m$ and the pushforwards of the conditional measures of $\mu$ on the unstable and stable leaves of the Teichmuller flow, whose spherical averages against $s_X$ give $\Lambda$. Here $\Lambda(X)=\nu(\{\beta\in\mathcal{ML}_g:\mathrm{Ext}_\beta(X)\le 1\})$ is the Thurston volume of the extremal-length unit ball, identified in the source paper with the leading coefficient of the counting function for multicurves of bounded extremal length, and proved bounded. The proof writes $m$ in polar coordinates, $dm(\pi(g_tq))=\Delta(q,t)\,ds_X(q)\,dt$, and establishes $\Delta(q,t)=e^{ht}\lambda^-(q)\lambda^+(g_tq)+O(e^{(h-\alpha)t})$ (Proposition 2.5) only under the condition that the geodesic segment $\{g_sq\}_{0\le s\le t}$ spends at least half its time in a fixed compact subset of the principal stratum; the exponent $\alpha>0$ depends on that compact, and without the recurrence condition only $|\Delta(q,t)|\le Ce^{ht}$ is available. Theorem 2.2 shows $s_X$ gives zero weight to the multiple zero locus, and Theorems 2.6 and 2.7 show that multiple-zero and non-recurrent directions are negligible at leading order; the comparison constants between the modified Hodge norm (the usual Hodge $L^2$-norm on differentials augmented by the values of the differential at the zeros, measured in the flat metric) and the Teichmuller and Euclidean metrics on leaves (Theorems 3.6 and 3.10) depend on compact subsets.

Two later developments are essential context. First, Dumas, as an application of a stratified Kahler metric on $Q(X)$ symplectomorphic to $\mathcal{ML}_g$, recorded an argument of Mirzakhani proving that $\Lambda$ is constant; denoting its value by $\Lambda_g>0$, the leading constants above become universal, e.g. $|\mathrm{Mod}_g\cdot Y\cap B_R(X)|\sim\frac{\Lambda_g^2}{h\,m(\mathcal{M}_g)}\,e^{hR}$. Second, Arana-Herrera made the asymptotics effective: for $X,Y$ in a compact subset $K\subset\mathcal{T}_g$, $\#\{\gamma\in\mathrm{Mod}_g:\gamma\cdot Y\in B_R(X)\}=\frac{\Lambda_g^2}{h\,m(\mathcal{M}_g)}\,e^{hR}+O_K(e^{(h-\kappa)R})$ with $\kappa=\kappa(g)>0$, together with effective sector and bisector counts, an effective polar-coordinates estimate valid throughout the principal stratum with explicit dependence on the length of the shortest saddle connection (a saddle connection is a flat geodesic segment joining two zeros of $q$ with no zero in its interior), and effective control of contributions of segments spending much of their time in the thin part (the region of the stratum where the flat length of some curve becomes small), building on thin-part large-deviation estimates of Eskin-Mirzakhani-Rafi and Hodge-Teichmuller norm comparisons of Kahn-Wright. The effective constants depend on the compact $K$, the exponents are not sharp, and the centers are restricted to compacts. The open problem below asks for the sharp, uniform completion of this quantitative theory.

Problem Statement

Let $g\ge 2$, let $\mathcal{T}_g$ be Teichmuller space of closed genus-$g$ surfaces with the Teichmuller metric, $B_R(X)\subset\mathcal{T}_g$ the ball of radius $R$, $\mathrm{Mod}_g$ the mapping class group, $h=6g-6$, $m$ the pushforward of the Masur-Veech measure from the unit-area quadratic differential bundle, $\mathcal{M}_g=\mathcal{T}_g/\mathrm{Mod}_g$ the moduli space (with $m(\mathcal{M}_g)<\infty$), and $\Lambda_g$ the Hubbard-Masur constant, defined for any $X\in\mathcal{T}_g$ by $\Lambda_g=\nu(\{\beta\in\mathcal{ML}_g:\mathrm{Ext}_\beta(X)\le 1\})$ and independent of $X$. Write $F_R(X,Y)=\#\{\gamma\in\mathrm{Mod}_g:\gamma\cdot Y\in B_R(X)\}$. Known: $F_R(X,Y)\sim\frac{\Lambda_g^2}{h\,m(\mathcal{M}_g)}e^{hR}$ with sector, bisector, and ball-volume analogues, and the effective estimate $F_R(X,Y)=\frac{\Lambda_g^2}{h\,m(\mathcal{M}_g)}e^{hR}+O_K(e^{(h-\kappa)R})$ for $X,Y$ in a compact $K\subset\mathcal{T}_g$ with a non-sharp $\kappa=\kappa(g)>0$. Resolve the following, which the source paper's theorem-level limitations identify (no error terms in its main asymptotic theorems, a polar-coordinates estimate valid only under a recurrence condition with a compact-dependent exponent, treatment of the multiple zero locus only at leading order, and compact-dependent metric-comparison constants) and which the later effective literature leaves open:

(A) Optimal error terms and secondary structure. Determine, for centers $X,Y$ ranging over a compact set $K\subset\mathcal{T}_g$ (the regime of the known effective estimate; the extension to arbitrary centers is item (B)), the optimal (sharp, two-sided) decay exponent for the error $F_R(X,Y)-\frac{\Lambda_g^2}{h\,m(\mathcal{M}_g)}e^{hR}$, and decide whether a full asymptotic expansion with identifiable secondary terms exists; do the same for the sector and bisector counts and for the volume growth $m(B_R(X))$.

(B) Uniformity in the centers. Extend the effective estimates to arbitrary $X,Y\in\mathcal{T}_g$, with an explicit description of how the error constants degenerate as $X$ or $Y$ leaves every compact set (pinched curves, collapsing flat cylinders, the Deligne-Mumford boundary of nodal surfaces), and determine the sharp dependence of the decay exponent in the polar-coordinates estimate on the recurrence data (the chosen compact subset of the principal stratum), replacing the current sufficient bounds for geodesic segments violating the recurrence condition.

(C) Subleading contribution of the multiple zero locus. Determine the sharp rate at which neighborhoods of the multiple zero locus contribute to the error terms in the counting and volume asymptotics, and the sharp asymptotic behavior of the relevant metric-comparison constants as one approaches higher-multiplicity strata, beyond the effective smallness bounds currently available.

References

1. Jayadev Athreya, Alexander Bufetov, Alex Eskin, Maryam Mirzakhani, 'Lattice point asymptotics and volume growth on Teichmüller space', Duke Mathematical Journal 161 (2012), no. 6, 1055-1111, DOI 10.1215/00127094-1548443 (preprint arXiv:math/0610715), https://doi.org/10.1215/00127094-1548443
2. David Dumas, 'Skinning maps are finite-to-one', Acta Mathematica 215 (2015), no. 1, 55-126, DOI 10.1007/s11511-015-0129-6 (preprint arXiv:1203.0273), https://doi.org/10.1007/s11511-015-0129-6
3. Francisco Arana-Herrera, 'Effective mapping class group dynamics, I: Counting lattice points in Teichmüller space', Duke Mathematical Journal 172 (2023), no. 8, 1437-1529, DOI 10.1215/00127094-2022-0066 (preprint arXiv:2010.03123), https://doi.org/10.1215/00127094-2022-0066
4. Alex Eskin, Maryam Mirzakhani, Kasra Rafi, 'Counting closed geodesics in strata', Inventiones Mathematicae 215 (2019), no. 2, 535-607, DOI 10.1007/s00222-018-0832-y (preprint arXiv:1206.5574), https://doi.org/10.1007/s00222-018-0832-y
5. Jeremy Kahn, Alex Wright, 'Hodge and Teichmüller', Journal of Modern Dynamics 18 (2022), 149-160, DOI 10.3934/jmd.2022007 (preprint arXiv:2106.04665), https://doi.org/10.3934/jmd.2022007
6. Weixu Su, Shenxing Zhang, 'Volume of unit balls associated to quadratic differentials', arXiv:2510.21365 (2025), https://arxiv.org/abs/2510.21365
7. Spencer Dowdall, Howard Masur, 'Counting mapping classes by Nielsen-Thurston type', arXiv:2305.03721 (v1 2023; revised v2, March 2026, carries this title and the expanded results), https://arxiv.org/abs/2305.03721