Circumcenter extension of Moebius maps to CAT(-1) spaces
Annales de l'Institut Fourier, Volume 74 (2024) no. 1, pp. 235-255.

Given a Moebius homeomorphism f:XY between boundaries of proper, geodesically complete CAT(-1) spaces X,Y, we describe an extension f ^:XY of f, called the circumcenter map of f, which is constructed using circumcenters of expanding sets. The extension f ^ is shown to coincide with the (1,log2)-quasi-isometric extension constructed in a previous paper of the author, and is locally 1/2-Holder continuous. When X,Y are complete, simply connected manifolds with sectional curvatures K satisfying -b 2 K-1 for some b1 then the extension f ^:XY is a (1,(1-1 b)log2)-quasi-isometry, and is surjective. Circumcenter extension of Moebius maps is natural with respect to composition with isometries.

Soit X,Y des CAT(-1) espaces propres et géoésiquement complètes, et soit f:XY un homéomorphisme qui préserve le birapport. Nous décrivons une extension f ^:XY de f, que nous appelons l’extension circoncentre de f, et qu’on construit à l’aide des circoncentres des ensembles agrandissants. On montre que l’extension f ^ coincide avec l’extension (1,log2)-quasi-isométrique déjà construit par l’auteur dans un article précédent. Quand X,Y sont des variétés riemanniennes simplement connexes à courbures dans l’interval [-b 2 ,-1], l’extension f ^:XY est un (1,(1-1 b)log2)-quasi-isométrie surjectif. L’extension circoncentre est naturelle par rapport à composition avec des isométries.

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DOI: 10.5802/aif.3582
Classification: 51F30
Keywords: $\mathrm{CAT}(-1)$ space, Moebius map, circumcenter.
Mot clés : $\mathrm{CAT}(-1)$ espace, birapport, circoncentre.
Biswas, Kingshook 1

1 Indian Statistical Institute Kolkata (India)
License: CC-BY-ND 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights
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Biswas, Kingshook. Circumcenter extension of Moebius maps to $\mathrm{CAT}(-1)$ spaces. Annales de l'Institut Fourier, Volume 74 (2024) no. 1, pp. 235-255. doi : 10.5802/aif.3582. https://aif.centre-mersenne.org/articles/10.5802/aif.3582/

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