# zbMATH — the first resource for mathematics

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
Full Text:
##### References:
 [1] Ballmann W.,Lectures on spaces of nonpositive curvature, Birkhäuser, 1995. · Zbl 0834.53003 [2] Beardon A., Maskit B.,Limit points of Kleinian groups and finite sided fundamental polyhedra, Acta Math.,132, 1–12 (1974). · Zbl 0277.30017 [3] Benedetti R., Petronio C.,Lectures on hyperbolic geometry, Universitext, Springer-Verlag, 1992. · Zbl 0768.51018 [4] Bridson M. R., Haefliger A.,Metric spaces of non-positive curvature, Grundlehren der Mathematischen Wissenschaften,319 (1999). Berlin, Springer. · Zbl 0988.53001 [5] Charitos C.,Closed geodesics in ideal polyhedra of dimension 2, Rocky Mountain Jour. of Math.,26 2 (1996), p. 507–521. · Zbl 0869.52003 [6] Charitos C., Papadopoulos A.,The geometry of ideal 2-dimensional simplicial complexes, Glasgow Math. J.,43 (2001), 39–66. · Zbl 0977.57003 [7] Charitos C., Tsapogas G.,Complexity of geodesics on 2-dimensional ideal polyhedra and isotopies, Math. Proc. Camb. Phil. Soc.,121 (1997), 343–258. · Zbl 0890.57047 [8] Charitos C., Tsapogas G.,Geodesic flow on ideal polyhedra, Can. J. Math.,49 (4) (1997), 696–707. · Zbl 0904.52004 [9] Coornaert M.,Sur les groupes proprement discontinus d’isometries des espaces hyperboliques au sens de Gromov, Thèse U.L.P., Publication de l’IRMA, 1990. · Zbl 0777.53044 [10] Epstein D. B. A., Penner R. C.,Euclidean decompositions of noncompact hyperbolic manifolds, J. Differ. Geom.,27 (1988), 67–80. · Zbl 0611.53036 [11] Ghys E., de la Harpe P., (Ed.)Sur les groupes hyperboliques d’après Mikhael Gromov, Progress in Mathematics,83. Birkhäuser Boston, Inc., Boston, MA, 1990. · Zbl 0731.20025 [12] Gromov M.,Hyperbolic groups, in Essays Croup Theory, MSRI Publ.,8 Springer, 1987. · Zbl 0634.20015 [13] Gromov M.,Structures métriques pour les variétés riemanniennes (rédigé par J. Lafontaine et P. Pansu), Fernand Nathan, Paris, 1981. [14] Paulin F.,Construction of hyperbolic groups via hyperbolizations of polyhedra, in Group Theory from a Geometrical Viewpoint, Proceedings of a conference in Trieste, World Scientific, 1991. · Zbl 0843.20032 [15] Petronio C.,Ideal triangulations of hyperbolic 3-manifolds, Proceedings of the 16th national congress of the Italian Mathematical Society, Napoli, 1999, Bologna, Unione Matematica Italiana, 593–608 (2000). · Zbl 0983.57013 [16] Ratcliffe J.,Foundations of Hyperbolic Manifolds, Springer-Verlag, 1994. · Zbl 0809.51001 [17] Thurston W. P.,Three-dimensional Geometry and Topology, Princeton U. Press,1 (1977).
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.