Self-representations of the Möbius group
Annales Henri Lebesgue, Volume 2 (2019), pp. 259-280.

Keywords Möbius group, Lobatchevsky space, hyperbolic space, infinite-dimensional space

### Abstract

Contrary to the finite-dimensional case, the Möbius group admits interesting self-representations when infinite-dimensional. We construct and classify all these self-representations.

The proofs are obtained in the equivalent setting of isometries of Lobachevsky spaces and use kernels of hyperbolic type, in analogy with the classical concepts of kernels of positive and negative type.

### References

[BdlHV08] Bekka, Bachir; de la Harpe, Pierre; Valette, Alain Kazhdan’s property (T), New Mathematical Monographs, 11, Cambridge University Press, 2008, xiv+472 pages | MR | Zbl

[Bek03] Bekka, Bachir Kazhdan’s property (T) for the unitary group of a separable Hilbert space, Geom. Funct. Anal., Volume 13 (2003) no. 3, pp. 509-520 | DOI | MR | Zbl

[BGS85] Ballmann, Werner; Gromov, Mikhail; Schroeder, Viktor Manifolds of nonpositive curvature, Progress in Mathematics, 61, Birkhäuser, 1985, vi+263 pages | MR | Zbl

[BH99] Bridson, Martin Robert; Haefliger, André Metric spaces of non-positive curvature, Grundlehren der Mathematischen Wissenschaften, 319, Springer, 1999, xxii+643 pages | MR | Zbl

[BIM05] Burger, Marc; Iozzi, Alessandra; Monod, Nicolas Equivariant embeddings of trees into hyperbolic spaces, Int. Math. Res. Not. (2005) no. 22, pp. 1331-1369 | DOI | MR | Zbl

[Can11] Cantat, Serge Sur les groupes de transformations birationnelles des surfaces, Ann. Math., Volume 174 (2011) no. 1, pp. 299-340 | DOI | MR | Zbl

[Gan66] Gans, David A new model of the hyperbolic plane, Am. Math. Mon., Volume 73 (1966) no. 3, pp. 291-295 | DOI | MR | Zbl

[Gro01] Gromov, Mikhail $\mathrm{CAT}\left(\kappa \right)$-spaces: construction and concentration, Zap. Nauchn. Semin. (POMI), Volume 280 (2001), pp. 101-140 | Zbl

[Kes51] Kestelman, Hyman Automorphisms of the field of complex numbers, Proc. Lond. Math. Soc., Volume 53 (1951), pp. 1-12 | DOI | MR | Zbl

[Leb07] Lebesgue, Henri Sur les transformations ponctuelles, transformant les plans en plans, qu’on peut définir par des procédés analytiques (Existrait d’une lettre adressée à M. C. Segre), Torino Atti, Volume 42 (1907), pp. 532-539 | Zbl

[Man86] Manin, Yuri Cubic forms. Algebra, geometry, arithmetic, North-Holland Mathematical Library, 4, North-Holland Publishing Co., Amsterdam, 1986, x+326 pages (translated from the Russian by M. Hazewinkel)

[Mon18] Monod, Nicolas Notes on functions of hyperbolic type (2018) (https://arxiv.org/abs/1807.04157v1)

[MP14] Monod, Nicolas; Py, Pierre An exotic deformation of the hyperbolic space, Am. J. Math., Volume 136 (2014) no. 5, pp. 1249-1299 | DOI | MR | Zbl

[Res89] Reshetnyak, Yurii G. Space mappings with bounded distortion, Translations of Mathematical Monographs, 73, American Mathematical Society, 1989, xvi+362 pages (translated from the Russian by H. H. McFaden) | MR | Zbl

[RR07] Ricard, Éric; Rosendal, Christian On the algebraic structure of the unitary group, Collect. Math., Volume 58 (2007) no. 2, pp. 181-192 | MR | Zbl

[SSV12] Schilling, René L.; Song, Renming; Vondraček, Zoran Bernstein functions. Theory and applications, De Gruyter Studies in Mathematics, 37, Walter de Gruyter, 2012, xiv+410 pages | Zbl

[Tsa13] Tsankov, Todor Automatic continuity for the unitary group, Proc. Am. Math. Soc., Volume 141 (2013) no. 10, pp. 3673-3680 | DOI | MR | Zbl