We prove an extension of Basmajian's identity to n-Hitchin representations of compact bordered surfaces. For n = 3, we show that this identity has a geometric interpretation for convex real projective structures analogous to Basmajian's original result. As part of our proof, we demonstrate that, with respect to the Lebesgue measure on the Frenet curve associated to a Hitchin representation, the limit set of an incompressible subsurface of a closed surface has measure zero. This generalizes a classical result in hyperbolic geometry. Finally, we recall the Labourie-McShane extension of the McShane-Mirzakhani identity to Hitchin representations and note a close connection to Basmajian's identity in both the hyperbolic and the Hitchin settings.

Sullivan showed that there exists K0 such that if Ω⊂double-struckC^ is a simply connected hyperbolic domain, then there exists a conformally natural K0‐quasiconformal map from normalΩ to the boundary Dome(Ω) of the convex hull of its complement which extends to the identity on ∂Ω. Explicit upper and lower bounds on K0 were obtained by Epstein, Marden, Markovic and Bishop. We improve on these bounds, by showing that one may choose K0⩽7.1695.

Let S$S$ be a surface of negative Euler characteristic and consider a finite filling collection Γ$\Gamma$ of closed curves on S$S$ in minimal position. An observation of Foulon and Hasselblatt shows that PT(S)∖trueΓ̂$PT(S) \setminus \widehat {\Gamma }$ is a finite‐volume hyperbolic 3‐manifold, where PTfalse(Sfalse)$PT(S)$ is the projectivized tangent bundle and normalΓ̂$\widehat \Gamma$ is the set of tangent lines to Γ$\Gamma$. In particular, volfalse(PT(S)∖trueΓ̂false)$vol(PT(S) \setminus \widehat {\Gamma })$ is a mapping class group invariant of the collection Γ$\Gamma$. When Γ$\Gamma$ is a filling pair of simple closed curves, we show that this volume is coarsely comparable to Weil–Petersson distance between strata in Teichmüller space. Our main tool is the study of stratified hyperbolic links normalΓ¯$\overline{\Gamma }$ in a Seifert‐fibered space N$N$ over S$S$. For such links, the volume of N∖Γ¯$N\setminus \overline{\Gamma }$ is coarsely comparable to expressions involving distances in the pants graph.

We consider circle packings and, more generally, Delaunay circle patternsarrangements of circles arising from a Delaunay decomposition of a finite set of points -on surfaces equipped with a complex projective structure. Motivated by a conjecture of Kojima, Mizushima and Tan, we prove that the forgetful map sending a complex projective structure admitting a circle packing with given nerve (resp. a Delaunay circle pattern with given nerve and intersection angles) to the underlying complex structure is proper.

Let Γ be a hyperbolic link in a Seifert-fibered space N over a surface S of negative Euler characteristic. When Γ is stratified, as defined in this paper, we show that the volume of N \ Γ is quasi-isometric to expressions involving distances in the pants graph. When S is a punctured torus or a four punctured sphere and N = P T 1 (S), we show that the canonical lift Γ of a filling collection Γ of essential simple closed curves is always stratified and that the volume is quasi-isometric to curve complex distance. Lastly, we given large families of stratified hyperbolic canonical links in P T 1 (S).