6 Equal parameters and the torus–sphere correspondence

The familiar punctured correspondence extends geometrically to equal positive boundaries. Let $Q(c^4)$ be a four-holed sphere with all four boundary lengths $c$, and let $T(2c)$ be a one-holed torus with boundary length $2c$. The relevant construction is through a common orbifold: $$T(2c)\xrightarrow{\;2:1\;} O(c;\pi,\pi,\pi) \xleftarrow{\;4:1\;}Q(c,c,c,c).$$(6.1) The middle surface is a sphere with one geodesic boundary and three angle-$\pi$ orbifold points. The torus quotient uses its elliptic involution; the sphere quotient uses its Klein-four symmetry. This works for every $c>0$ and limits to the cusped case [63, Section 3].

A matching simple curve on $T$ covers an arc between two orbifold points twice; its counterpart on $Q$ covers that arc four times. Thus $$\ell_Q=2\ell_T.$$(6.2) The construction gives a correspondence of simple-curve classes, not only an agreement of dimension or of volume polynomials.

WP forms scale by the degree of a local-isometric covering on a family of lifted metrics. This follows from naturality of the Goldman symplectic pairing: its surface integral is multiplied by the covering degree. On the common marked orbifold family, $$\omega_T=2\omega_O,\qquad \omega_Q=4\omega_O, \qquad\text{so}\qquad\omega_Q=2\omega_T.$$ This comparison takes place before dividing by mapping classes. Branch points are cone points of the quotient, so the maps are local isometries in the orbifold sense; their orders $2$ and $4$ multiply the same integral pairing. The statement does not involve varying the boundary length $c$. Generically the equal-boundary sphere has its Klein-four isometry group. There are $4!/4=6$ inequivalent ways to label its boundaries. The torus has its generic elliptic involution, which acts ineffectively on Teichmüller space. The Klein-four transformations permute the four sphere boundaries freely and transitively, so none survives after all four labels are fixed. The generic degree from labeled spheres to the effective torus moduli space is therefore $6$, and the local area ratio is $2$. Consequently $$\boxed{ V_{0,4}(c,c,c,c)=12V_{1,1}^{\mathrm{eff}}(2c) =24V_{1,1}^{\mathrm{stack}}(2c).}$$(6.3) Indeed, $24((2c)^2+4\pi^2)/48=2\pi^2+2c^2$, as required by (1.2). Exceptional extra automorphisms are accounted for by orbifold integration. Forgetting the ineffective torus involution is the source of a common factor-of-two discrepancy.

The cone analogue can be constructed directly. For $0<\phi<\pi$, the elliptic quotient of a torus with one cone opening $2\phi$ is a sphere with openings $(\phi,\pi,\pi,\pi)$. Its Klein-four cover branched over the three angle-$\pi$ points has four openings equal to $\phi$. Uniqueness of conical uniformization makes the conformal symmetries isometries. The same degree and labeling argument gives $$V_{0,4}(i\phi,i\phi,i\phi,i\phi) =2\pi^2-2\phi^2=24V_{1,1}(2i\phi).$$(6.4) The common quotient cone surface has area $\pi-\phi$, so $\phi<\pi$ is exactly the hyperbolic range. There are no collision corrections in this equal-cone sphere chamber. The torus has one cone opening $2\phi<2\pi$ and no other noncuspidal ends, so its polynomial evaluation is covered by the single-cone regime of Section 5. Thus both sides of (6.4) are geometric volumes in the stated interval. The endpoint $\phi=\pi$ has zero area and is a limit, not an additional hyperbolic moduli space.

This correspondence does not assert a direct unbranched degree-two map from the four-holed sphere to the one-holed torus. The common orbifold and its two different covering degrees are the precise geometric statement.