The Margulis lemma and the thick and thin decomposition for convex real projective surfaces (Q1924007)

Let \(S^2\) be the standard unit sphere and let \(\Omega\) be a simply convex domain of an open hemisphere of \(S^2\). If \(x,y\) are two distinct interior points of \(\Omega,q\) the geodesic passing through \(x\) and \(y\) and \(x_\infty\), \(y_\infty\) the points of intersection of \(q\) with the boundary of \(\Omega\) \((x\) being between \(x_\infty\) and \(y)\), then the expression \[ d^H_{\text{int }\Omega} (x,y) = \lg (x_\infty, y_\infty, y,x) \] (where \((x_\infty, y_\infty, y,x)\) is the cross-ratio) defines the Hilbert metric. Let \(\Gamma\) be a projective automorphism of \(S^2 \) acting properly discontinuously and freely on \(\Omega\), so that \(\Sigma= \Omega/ \Gamma\) is a differentiable surface. Since \(\Omega\) has an induced projective structure from \(S^2\), \(\Sigma\) has a projective structure. Since \(\Gamma\) acts as isometries with respect to \(d^H_{\text{int }\Omega}\), the interior \(\Sigma^0\) of \(\Sigma\) has a Hilbert metric induced from this which is denoted by \(d^H_{\Sigma^0}\). The author proves the following Theorem: Let \(x\in \text{int} \Omega\). There exists an independent constant \(\varepsilon\), \(0< \varepsilon <1\), such that for \(0<\delta<\varepsilon\) the subgroup \(\Gamma_\delta(x)\) of \(\Gamma\) generated by elements \(\gamma\) with \(d^H_{\text{int }\Omega} (x, \gamma (x)) \leq \delta\) is an infinite cyclic subgroup or a trivial group. Given \(\varepsilon >0\), the \(\varepsilon\)-thin part of \(\Sigma^0 \) is the set of points of \(\Sigma^0\) through which closed curves of \(d^H_{\Sigma^0}\)-length less or equal to \(\varepsilon\) pass. The \(\varepsilon\)-thick part of \(\Sigma^0\) is the complement in \(\Sigma^0\) of the \(\varepsilon\)-thin part. The author investigates some topological properties of the decomposition of \(\Sigma^0\) into the \(\varepsilon\)-thin and \(\varepsilon\)-thick parts.