The Thurston metric on the punctured-torus plane

Definitions and mathematical statement

Identify the marked complete finite-area once-punctured-torus Teichmüller space $\mathcal T(S_{1,1})$ with the upper half-plane as a parameter space. The question asks for an explicit description of its hyperbolic-surface Thurston metric, not the Poincaré metric of that half-plane or a metric on flat tori.

The supplied aid uses positive Fricke coordinates $(x,y,z)$, the trace coordinates of a fixed intersecting curve basis and its Farey neighbor, on \[ x^2+y^2+z^2=xyz,\qquad x,y,z>2. \] Primitive slopes $r\in{\mathbb{Q}}\cup\{\infty\}$, with primitive integer pairs identified up to sign, index essential simple closed curves. Let $T_r$ be their positive trace functions, initialized by $x,y,z$ on a Farey triangle. For adjacent triangles sharing the two slopes with functions $T_a,T_b$, the opposite functions satisfy \[ T_c+T_{c'}=T_aT_b. \] The reported Markoff-tree formulas are \[ \ell_r(X)=2\operatorname{arcosh}(T_r(X)/2),\qquad d_{\rm Th}(X,Y)=\log\sup_r \frac{\operatorname{arcosh}(T_r(Y)/2)}{\operatorname{arcosh}(T_r(X)/2)}. \] This is an exact supremum over all primitive slopes, not a claim that finitely many recursive steps always certify its value. Matching Fricke and upper-half-plane markings is necessary; the coordinate dictionary is not supplied here.

For a tangent vector $v$ to the Fricke surface the corresponding reported infinitesimal formula is \[ \|v\|_{\rm Th,X}=\sup_r \frac{({\,\mathrm{d}} T_r)_X(v)}{\sqrt{T_r(X)^2-4}\,\operatorname{arcosh}(T_r(X)/2)}. \] The derivative covectors are obtained from the same recursion, \[ {\,\mathrm{d}} T_{c'}=T_b\,{\,\mathrm{d}} T_a+T_a\,{\,\mathrm{d}} T_b-{\,\mathrm{d}} T_c. \] No absolute value is inserted in the asymmetric infinitesimal norm.

The aid gives an equivalent stable-length description. Let $\|\cdot\|_X$ be the McShane--Rivin length norm on $H_1(S_{1,1};{\mathbb{R}})$, normalized so that a primitive integral class has the length of its simple geodesic. Then \[ e^{d_{\rm Th}(X,Y)}=\sup_{v\ne0}\frac{\|v\|_Y}{\|v\|_X}, \] the directed operator norm of the identity from the $X$-normed plane to the $Y$-normed plane. It also reports known Fenchel--Nielsen formulas for the two stretch paths associated to the two complete-lamination choices for a simple closed curve, giving exact unit-speed directed geodesics. Their explicit twist formulas and conventions are not supplied. These descriptions show why the asymmetric Thurston metric is different from the symmetric Poincaré metric on the same parameter space, without claiming a finite closed formula for the maximizing slope in every pair.

Short English statement

Describe the punctured-torus Thurston metric through recursively generated trace lengths, and its tangent norm through the derivatives of the same traces. The supplied stable-length formulation is a directed operator norm; it is distinct from both Poincaré geometry and the cited flat-torus setting.

Sources

• Problem 5.1, AIM workshop Lipschitz metric on Teichmueller space, https://aimath.org/pastworkshops/lipschitzteichproblems.pdf. The source mentions Belkhirat, Papadopoulos and Troyanov's study of flat-torus Thurston metrics, without a full reference.
• The research summary names McShane--Rivin stable length norms and known Fenchel--Nielsen stretch-path formulas; full titles, URLs and normalization details are not supplied.
• Local UnsolvedMath registry, permanent ID 20003119, exact code catalogue entry; canonical location aim-topology-notes.json, notes[206], dated 14 August 2026; statement labeled exact.
• Research attempt 1: partial result, problem partially solved at the run, rigor label proved, novelty confidence low; registry status partially solved. The Markoff-recursion and norm package is supplied as a machine-generated research aid, not peer reviewed; independent verification is required before citation.