17 Unfolding and the problem of finding constant series

17.1 What a volume computation determines

Let $\Gamma$ be the mapping class group of an oriented finite-type surface with fixed boundary lengths, and let $\gamma$ be a fixed nonperipheral curve or multicurve class. Use the same WP orbifold measure on both quotients. For a measurable nonnegative function, or one for which $\int_{\Gamma\backslash\mathcal T}\sum_\delta|F(\ell_\delta)|\,\mathop{}\!\mathrm dV_{\mathrm{WP}}<\infty$, the basic unfolding identity is $$\int_{\Gamma\backslash\mathcal T} \sum_{\delta\in\Gamma\gamma}F(\ell_\delta(X))\,\mathop{}\!\mathrm dV_{\mathrm{WP}}(X) =\int_{\mathop{\mathrm{Stab}}(\gamma)\backslash\mathcal T} F(\ell_\gamma(X))\,\mathop{}\!\mathrm dV_{\mathrm{WP}}(X).$$(17.1) The orbit contains each class once; $\mathop{\mathrm{Stab}}(\gamma)$ preserves the entire chosen decoration, including any orientation, ordering or component weights. Integration over coset translates of a fundamental domain proves the formula, with ineffective isotropy retained on both sides. For nonnegative functions either side may be infinite. Simplicity of the orbit curves is not required.

Simplicity becomes important when evaluating the right side. If $\gamma$ is a disjoint simple multicurve, cut along it and use Fenchel–Nielsen coordinates. Each twist period integrates to $\ell_j$, so the resulting integral has factors $\prod_j\ell_j\,\mathop{}\!\mathrm d\ell_j$ times WP volumes of the components, with the relevant symmetry factors. This is the geometric integration mechanism underlying Mirzakhani’s volume recursion [75, Theorem 7.1 and Lemma 7.3]. The symmetry and one-handle factors in that formula must be converted using the volume convention for each component; they are not extra factors to append to an already stack-normalized quotient.

Suppose a series $S(X)$ is independently known to be constant, say $C$, and $0<V_{g,n}<\infty$ in the chosen normalization. Then $$C=\frac{1}{V_{g,n}} \int_{\mathcal M_{g,n}}S(X)\,\mathop{}\!\mathrm dV_{\mathrm{WP}}(X),$$(17.2) and unfolding can calculate its value. If constancy has not been proved, the same quotient computes only its average. This is the precise logical distinction behind the proposal to recover an identity from a volume.

On the punctured torus, for an unoriented essential simple curve orbit, the stack-normalized formula is $$\int_{\mathcal M_{1,1}}\sum_{\alpha\text{ simple}}f(\ell_\alpha)\,\mathop{}\!\mathrm dV_{\mathrm{WP}} =\frac12\int_0^\infty x f(x)\,\mathop{}\!\mathrm dx.$$(17.3) Indeed, the effective stabilizer quotient has coordinates $x>0$ and $0\leq\tau<x$, while the ineffective torus involution contributes the single factor $1/2$. This is the stack convention of Section 6; it is not a second quotient by cuff exchange. For $f(x)=(1+e^x)^{-1}$, expand in $e^{-x}$ to obtain $$\int_0^\infty\frac{x\,\mathop{}\!\mathrm dx}{1+e^x} =\sum_{m\ge1}\frac{(-1)^{m-1}}{m^2}=\frac{\pi^2}{12}.$$ Termwise integration is justified by $\sum_m\int_0^\infty xe^{-mx}\,\mathop{}\!\mathrm dx=\sum_m m^{-2}<\infty$. For a finite numerical check, the integral beyond $R$ is at most $(R+1)e^{-R}$, and the alternating sum through $m=M$ has error at most $(M+1)^{-2}$. The integral of the series is $\pi^2/24$; division by $V_{1,1}=\pi^2/12$ gives $C=1/2$. Its pointwise constancy is McShane’s theorem [73]: $$\sum_{\alpha\text{ simple, unoriented}} \frac{1}{1+e^{\ell_\alpha(X)}}=\frac12.$$(17.4) The tetrahedral volume calculation and the covering argument provide an independent route to the volume used in this normalization check.

17.2 Operations that preserve constancy

Finite linear combinations of constant series preserve constancy. Parameter integrals do too when the constant is integrable and Fubini’s theorem applies to the joint sum and integral, or when all terms are nonnegative and Tonelli’s theorem applies. For differentiation on an identified parameter family, one sufficient hypothesis is local uniform absolute convergence of both the series and its first derivatives. Uniform convergence of the original series alone is insufficient. Products give constant double series when Tonelli’s theorem or absolute convergence applies. These operations construct kernels together with their required convergence hypotheses.

Their geometric interpretation needs to be retained. A product usually indexes pairs of curves that may intersect; it is not automatically a disjoint-multicurve sum. A parameter derivative also differentiates lengths if the metric varies with that parameter. In particular, differentiating a constant series with respect to the metric gives the one-form identity $$\sum_\gamma F'(\ell_\gamma)\,\mathop{}\!\mathrm d\ell_\gamma=0,$$(17.5) under the stated local $C^1$ control, for a kernel with no additional explicit metric dependence. It does not generally give another scalar identity by differentiating only the displayed weight.

Two possible tests for a proposed constant series follow. One can try to show that (17.5) vanishes on a spanning set of tangent directions at every point; a $C^1$ series is then constant on each connected component. Alternatively, for real-valued $S\in L^2$ on a finite positive-volume moduli space, let $\overline S$ be its mean. Then $$\int_{\mathcal M}(S-\overline S)^2\,\mathop{}\!\mathrm dV_{\mathrm{WP}}=0$$(17.6) proves constancy almost everywhere, and everywhere if $S$ is continuous, since WP measure has full support. For a complex-valued series use $|S-\overline S|^2$ instead of the square. The second-moment calculation involves pairs of curves, including their intersection patterns. It can be substantially harder than the first unfolding integral.

Thus a useful search for further identities should ask for more than formal closure: a natural index set, a manageable positive weight, a geometric partition or telescoping mechanism, and a convergence proof. The constructions in Sections 18 and 20 illustrate these requirements.

17.3 Other established constant series

There are also constant series with substantially different kernels. For example, write $$\mathcal L(x)=\operatorname{Li}_2(x)+\tfrac12\log x\log(1-x)$$ for the Rogers dilogarithm on $(0,1)$, extended continuously to its endpoints, and define $$\begin{aligned} D(r)={}&\mathcal L\bigl(\mathop{\mathrm{sech}}^2(r/2)\bigr)\\ &+2\mathcal L\left(\frac{1+e^{-r}}2\right) -2\mathcal L\left(\frac{1-e^{-r}}2\right). \end{aligned}$$ On a complete finite-area once-punctured torus, Hu–Tan’s identity is $$\sum_{\alpha\text{ simple, unoriented}}D(\ell_\alpha)=\frac{\pi^2}{2}.$$(17.7) Here the index set is the nonperipheral simple spectrum with its curve multiplicities [63, Theorem 2, equation (2)]. Their Theorem 1 treats a positive geodesic boundary with a different, boundary-dependent kernel; the displayed $D$ is the cusped specialization.

For a closed oriented surface of genus $g\ge2$, Luo–Tan give explicit dilogarithmic weights $f(P),g(T)$ on properly embedded geometric pairs of pants and one-holed tori, each subsurface counted once, such that $$\sum_P f(P)+\sum_T g(T)=8\pi^2(g-1)=\operatorname{Vol}(T^1X).$$(17.8) Their construction partitions unit tangent directions up to a null set [70, Theorem 1.1 and Section 1.2]. Angular measure $2\pi$ on each unit tangent circle, together with area $4\pi(g-1)$, explains the normalization on the right. It illustrates another source of constant series: the invariant measure being partitioned need not live only on a horocycle or a geodesic boundary. The precise, longer subsurface weights are given in that reference.