\(2\pi\)-grafting and complex projective structures. I (Q906182)

Let \(S\) be a closed oriented surface of genus at least 2. Gallo, Kapovich and Marden asked in their paper [\textit{D. Gallo} et al., Ann. Math. (2) 151, No. 2, 625--704 (2000; Zbl 0977.30028)] the following question, which is called the ``Grafting conjecture'': Given two projective structures sharing (arbitrary) holonomy \(\rho: \pi_{1}(S) \rightarrow PSL(2,\mathbb C)\), is there a sequence of graftings and ungraftings that transforms one to the other? In the paper under review the author shows that this conjecture is true ``locally'' in the space \(\mathcal{GL}\) of geodesic laminations on \(S\). He first shows the following result: Let \(\rho: \pi_{1}(S) \rightarrow PSL(2,\mathbb {C})\) be a homomorphism. Suppose that there is a \(\rho\)-equivariant pleated surface \(\beta: {\mathbb H^{2}} \rightarrow {\mathbb H^{3}}\) realizing \((\tau,\lambda) \in {\mathcal{T} \times \mathcal{GL}}\), where \(\mathcal{T}\) is the space of marked hyperbolic structures on \(S\) (Teichmüller space). For every \(\epsilon > 0\), there exisits \(\delta > 0\) such that if there is another \(\rho\)-equivariant pleated surface \(\beta': {\mathbb H^{2}} \rightarrow {\mathbb H^{3}}\) realizing \((\sigma,\nu) \in {\mathcal{T} \times \mathcal{GL}}\) with \( \angle_{\tau}(\lambda, \nu) < \delta,\) then \(\sigma\) is \(\epsilon\)-close to \(\tau\) in \({\mathcal{T}}\) and \(\beta\) and \(\beta'\) are \(\epsilon\)-close. Let \(\mathcal{P}_{\rho}\) be the set of all projective structures with fixed holonomy \(\rho\). Let \(\mathcal{PML}\) be the space of projective measured laminations on \(S\). Applying the result above to pleated surfaces associated with projective structures, the author proves that if two projective structures in \(\mathcal{P}_{\rho}\) are close in \(\mathcal{PML}\) in Thurston coordinates, then they are related by a single grafting along a weighted multiloop. Let \(C \cong (\tau, L)\) be a projective structure on \(S\) with holonomy \(\rho\). Then there is a measured lamination \(L_{0}\) such that every closed leaf of \(L_{0}\) has weight less than \(2\pi\). Let \(C_{0}\) be the projective structure given by \((\tau, L_{0})\) in Thurston coordinates. The author obtains the stronger result: For every \(\epsilon > 0\) and every projective structure \(C \cong (\tau, L)\) on \(S\) with holonomy \(\rho\), there exists \(\delta > 0\), such that, if another projective structure \(C' \cong (\tau', L')\) with holonomy \(\rho\) satisfies \(\angle_{\tau}(L, L') < \delta\), then there are admissible train tracks on \({\mathcal T}_{0}, \mathcal{T}, \mathcal{T}'\) on \(C_{0}, C, C'\), respectively, that are isotopic on \(S\) and carry both \(L\) and \(L'\) (thus also \(L_{0}\)), so that: (1) \(C'\) is obtained by grafting \(C_{0}\) along a weighted multiloop \(M'\) carried by \({{\mathcal T}_{0}}\), such that \(M'\) is \(\epsilon\)-close to the measured lamination given by \(L'{}-L_{0}\) on \({\mathcal T}_{0}\). (2) If weighted multiloops \(M\) and \(M^\prime\) are carried by \(\mathcal{T}\) and \(\mathcal{T}'\), respectively, and \(M + M = {M^\prime} +M^\prime\) on the train tracks, then \(Gr_{M}(C)=Gr_{M^\prime}(C')\). This result gives a local characterization \(\mathcal{P}_{\rho}\) in \(\mathcal{GL}\) and generalizes the results on graftings obtained by Goldman, Kapovich and Ito.