Duality of the Kulkarni limit set for subgroups of \(\mathrm {PSL}(3,\mathbb {C})\)
Given a discrete subgroup \(\Gamma\) of PGL\((3,\mathbb{C})\), a general construction due to \textit{R. S. Kulkarni} [Math. Ann. 237, 253--272 (1978; Zbl 0369.20028)] gives an open set \(\Omega_K\) in \({\mathbf P}={\mathbf P}^2_{\mathbb{C}}\) where the action is proper. The complement \(\Lambda_K={\mathbf P}\setminus \Omega_K\) is called the Kulkarni limit set. The latter is obtained by taking \(L_0\cup L_1\cup L_2\) where \(L_0\) is the closure of points in \({\mathbf P}^2\) with infinite isotropy group, \(L_1\) is the closure of the set of accumulation points of orbits of points in \({\mathbf P}\setminus L_0\), and \(L_2\) is the closure of the set of accumulation points of orbits of compact subsets of \({\mathbf P}\setminus (L_0\cup L_1)\). For a large class of groups (e.g., when there are no global fixed point nor any globally invariant line), the open set \(\Omega_K\) is the unique maximal invariant open set where the action is proper. Inspired by work of \textit{J. P. Conze} and \textit{Y. Guivarc'h} [Sankhyā, Ser. A 62, No. 3, 367--385 (2000; Zbl 1115.37305)], the authors define another notion of limit set \(\widehat{L}(\Gamma)\), contained in the dual projective space \({\mathbf P}^*\) (a point in \({\mathbf P}^*\) is a line in \({\mathbf P}\), which of course induces a natural action of \(\Gamma\) on \({\mathbf P}^*\)). A point \({\mathbf q}\in{\mathbf P}\) is called a limit point of \(\Gamma\) if there exists an open set \(U\subset {\mathbf P}^*\) and a sequence \(g_n\) of pairwise distinct elements of \(\Gamma\) such that for every \({\mathbf p}\in U\), \(g_n({\mathbf p})\) converges to \({\mathbf q}\). The Conze-Guivarc'h limit set \(\widehat{L}(\Gamma)\) is the set of limit points in \({\mathbf P}^*\). The authors prove that, under the hypothesis that there is no global fixed point nor invariant line, the Kulkarni limit has an alternative description as the set of points in \({\mathbf P}\) lying on at least one element in the Conze-Guivarc'h limit set, each point in \(\widehat{L}(\Gamma)\subset {\mathbf P}^*\) now being seen as a subset of \({\mathbf P}\). They also show that for every closed subset \(C\subset {\mathbf P}^*\) and \(\varepsilon>0\), there exist a discrete subgroup \(\Gamma\) of PGL\((3,\mathbb{C})\) such that the Hausdorff distance between \(C\) and \(\widehat{L}(\Gamma)\) is smaller than \(\varepsilon\).
scientific article; zbMATH DE number 6932348