Geodesic
Boundary behaviour of automorphisms of a hyperbolic space
Trudy Matematicheskogo Instituta imeni V.A. Steklova, Tome 72 (2017) no. 4, pp. 645-670

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An automorphism of a Euclidean ball extends to a homeomorphic mapping of the closed ball even when the quasiconformality coefficient of the mapping increases unboundedly but in a controlled way upon approaching the boundary of the ball. By means of Poincaré's conformally Euclidean model of the Lobachevsky space, this yields a condition under which an automorphism of a hyperbolic space still extends to the ideal boundary (the absolute) of the space when translated into geometric language. Bibliography: 28 titles.
Keywords: hyperbolic space, Poincaré's model, quasiconformal mapping, equimorphism of the Lobachevsky space, asymptotic behaviour of the quasiconformality coefficient, boundary behaviour of a mapping. 
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     author = {V. A. Zorich},
     title = {Boundary behaviour of automorphisms of a hyperbolic space},
     journal = {Trudy Matematicheskogo Instituta imeni V.A. Steklova},
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     volume = {72},
     number = {4},
     year = {2017},
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     url = {http://geodesic.mathdoc.fr/item/RM_2017_72_4_a1/}
}
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V. A. Zorich. Boundary behaviour of automorphisms of a hyperbolic space. Trudy Matematicheskogo Instituta imeni V.A. Steklova, Tome 72 (2017) no. 4, pp. 645-670. http://geodesic.mathdoc.fr/item/RM_2017_72_4_a1/