Monod, Nicolas; Py, Pierre
Self-representations of the Möbius group
Annales Henri Lebesgue, Volume 2 (2019), p. 259-280

Metadata

KeywordsMöbius 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), Cambridge University Press, New Mathematical Monographs, Volume 11 (2008), xiv+472 pages

[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

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

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

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

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

[Gan66] Gans, David A new model of the hyperbolic plane, Am. Math. Mon., Volume 73 (1966) no. 3, pp. 291-295 | Zbl 0136.15101

[Gro01] Gromov, Mikhail CAT (κ)-spaces: construction and concentration, Zap. Nauchn. Semin. (POMI), Volume 280 (2001), pp. 101-140 | Zbl 1089.53029

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

[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 38.0096.05

[Man86] Manin, Yuri Cubic forms. Algebra, geometry, arithmetic, North-Holland Publishing Co., Amsterdam, North-Holland Mathematical Library, Volume 4 (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

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

[RR07] Ricard, Éric; Rosendal, Christian On the algebraic structure of the unitary group, Collect. Math., Volume 58 (2007) no. 2, pp. 181-192 | Zbl 1129.22012

[SSV12] Schilling, René L.; Song, Renming; Vondraček, Zoran Bernstein functions. Theory and applications, Walter de Gruyter, De Gruyter Studies in Mathematics, Volume 37 (2012), xiv+410 pages | Zbl 1257.33001

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