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On the isometries of ideal polyhedra. (English. French summary) Zbl 1194.57006
From the introduction: We prove that if \(X\) is a finite complete ideal polyhedron satisfying the local CAT\((-1)\) condition, then each isometry of \(X\) which is homotopic to the identity is the identity, provided that \(\pi_1 (X)\) is non-elementary. In the case \(n = 2\), the local CAT\((-1)\) condition is always satisfied. We prove also that the isometry group of \(X\) is finite. This result generalizes a theorem by A. F. Beardon and B. Maskit [Acta Math. 132, 1–12 (1974; Zbl 0277.30017)] concerning the isometries of complete hyperbolic \(n\)-manifolds.

MSC:
57M20 Two-dimensional complexes (manifolds) (MSC2010)
53C23 Global geometric and topological methods (à la Gromov); differential geometric analysis on metric spaces
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