Annex

• integer points
• non convexity

Pair of pants

• coprime \((p,q) \mapsto w_{p,q} \in \langle a,b \rangle\)
• extend length from \(\mathbb{Z}^2\) to \(\mathbb{R}^2\)
• technically, can be done in general using immersed train tracks

Goldman integer pts, orbifolds

\(x^2 + y^2 + z^2 - x y z = 20.\)

• different \(GL(2,\mathbb{Z})\) orbts
• (2,2,-2) three punctured sphere
• (0,3,1) orbifold, disc with 2 cone points
• (0, 4,2) orbifold, punctured disc one cone point

Three punctured sphere

Pair of pants

\(x^2 + y^2 + z^2 - x y z \geq 20.\)

(0,4,2) orbifold

(0,3,1) orbifold