Fenchel-Nielsen coordinates
/ FENG-kel NEEL-sun /
How do you write down a specific hyperbolic surface concretely enough to compute with? The trick is to cut it into the simplest possible pieces and record how the pieces were sized and reattached. Fenchel-Nielsen coordinates do exactly this: chop the surface along curves into 'pairs of pants' (three-holed spheres), then for each cutting curve list two numbers — how long the seam is, and how much you twisted before sewing it back. Those numbers, all together, name the surface uniquely and let you slide smoothly from one hyperbolic shape to any other.
Precisely, fix a pants decomposition of a closed genus-g surface S: a maximal collection of 3g-3 disjoint simple closed curves cutting S into 2g-2 pairs of pants. A pair of pants with prescribed boundary lengths has a UNIQUE hyperbolic structure (its three cuff lengths determine it rigidly), so the surface's geometry is captured by, for each of the 3g-3 curves, a length parameter l_i > 0 (the length of the unique geodesic in that curve's class) and a twist parameter tau_i in R (the signed shear applied when regluing the two pants across that curve). The map sending a marked hyperbolic surface to (l_1, ..., l_{3g-3}, tau_1, ..., tau_{3g-3}) is a real-analytic diffeomorphism T(S) -> (0,∞)^{3g-3} x R^{3g-3}.
So Fenchel-Nielsen coordinates give Teichmüller space a global, explicit coordinate system — a genuine atlas on the deformation space of hyperbolic structures, confirming dim T(S) = 6g-6. They are computationally central: the twist flows (increasing one tau_i at constant lengths) are exactly the 'twist deformations,' and Wolpert's beautiful formula says the Weil-Petersson symplectic form is simply the sum of d(l_i) ^ d(tau_i) — the lengths and twists are canonically conjugate, like positions and momenta. A subtle point: the twist parameter is genuinely real-valued (it does not wrap at 2pi), because twisting by the full curve length is a nontrivial Dehn twist, an element of the mapping class group, not a return to the same marked surface.
Take a one-holed torus (genus 1 with one boundary). A pants decomposition uses a single curve, giving 3g-3 = ... for the closed genus-2 case 3 curves; for the closed genus-2 surface specifically you get coordinates (l_1, l_2, l_3; tau_1, tau_2, tau_3) in (0,∞)^3 x R^3. Increase l_1 alone: the surface's first seam lengthens, fattening that handle. Increase tau_1 by exactly l_1 (one full twist) and you have applied a Dehn twist along curve 1 — the surface looks identical but its MARKING has changed, landing on a different point of T(S) in the same MCG-orbit.
Length and twist along each pants curve; one full-length twist is a Dehn twist, moving to a new marked structure.
Fenchel-Nielsen coordinates depend on the CHOICE of pants decomposition — different decompositions give different (real-analytically related) coordinate systems, not a canonical one. And the twist parameter is real, not an angle mod 2pi: a full twist is a Dehn twist, which genuinely changes the marking even though the underlying surface is unchanged.