Comptes Rendus
Théorie des groupes
Property FW and 1-dimensional piecewise groups
Comptes Rendus. Mathématique, Volume 359 (2021) no. 1, pp. 71-78.

Property FW is a natural combinatorial weakening of Kazhdan’s Property T. We prove that the group of piecewise homographic self-transformations of the real projective line, has “few” infinite subgroups with Property FW. In particular, no such subgroup is amenable or has Kazhdan’s Property T. These results are extracted from a longer paper. We provide a complete proof, whose main tools are the use of the notion of partial action and of one-dimensional geometric structures.

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DOI : 10.5802/crmath.155
Classification : 57S05, 57M50, 57M60, 20F65, 22F05, 53C10, 57S25
Yves Cornulier 1

1 CNRS and Univ Lyon, Univ Claude Bernard Lyon 1, Institut Camille Jordan, 43 blvd. du 11 novembre 1918, 69622 Villeurbanne, France.
Licence : CC-BY 4.0
Droits d'auteur : Les auteurs conservent leurs droits
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Yves Cornulier. Property FW and 1-dimensional piecewise groups. Comptes Rendus. Mathématique, Volume 359 (2021) no. 1, pp. 71-78. doi : 10.5802/crmath.155. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.155/

[1] Fernando Abadie Sobre ações parciais, fibrados de Fell e grupóides, Ph. D. Thesis, University of São Paulo, Brazil (1999)

[2] Fernando Abadie Enveloping actions and Takai duality for partial actions, J. Funct. Anal., Volume 197 (2003) no. 1, pp. 14-67 | DOI | MR | Zbl

[3] Bachir Bekka; Pierre de la Harpe; Alain Valette Kazhdan’s Property (T), New Mathematical Monographs, 11, Cambridge University Press, 2008 | MR | Zbl

[4] Dieter Betten Die Projektivitätengruppe der Moulton–Ebenen, J. Geom., Volume 13 (1979) no. 2, pp. 197-209 | DOI | MR | Zbl

[5] Dieter Betten; Alan Wagner Eine stückweise projektive topologische Gruppe im Zusammenhang mit den Moulton–Ebenen, Arch. Math., Volume 38 (1982) no. 3, pp. 280-285 | DOI | Zbl

[6] Yves Carrière; Étienne Ghys Relations d’équivalence moyennables sur les groupes de Lie, C. R. Math. Acad. Sci. Paris, Volume 300 (1985) no. 19, pp. 677-680 | Zbl

[7] Yves Cornulier Irreducible lattices, invariant means, and commensurating actions, Math. Z., Volume 279 (2015) no. 1-2, pp. 1-26 | DOI | MR | Zbl

[8] Yves Cornulier Group actions with commensurated subsets, wallings and cubings (2016) (https://arxiv.org/abs/1302.5982)

[9] Yves Cornulier Commensurating actions for groups of piecewise continuous transformations (2018) (https://arxiv.org/abs/1803.08572)

[10] Yves Cornulier Regularization of birational actions of FW groups (2019) (https://arxiv.org/abs/1910.07802 to appear in Confluentes Mathematici)

[11] François Dahmani; Koji Fujiwara; Vincent Guirardel Free groups of interval exchange transformations are rare, Groups Geom. Dyn., Volume 7 (2013) no. 4, pp. 883-910 | DOI | MR | Zbl

[12] Mikhail Ershov; Andrei Jaikin-Zapirain Property (T) for noncommutative universal lattices, Invent. Math., Volume 179 (2010) no. 2, pp. 303-347 | DOI | MR | Zbl

[13] Ruy Exel Partial actions of groups and actions of inverse semigroups, Proc. Am. Math. Soc., Volume 126 (1998) no. 12, pp. 3481-3494 | DOI | MR | Zbl

[14] Jacques Faraut; Khelifa Harzallah Distances hilbertiennes invariantes sur un espace homogène, Ann. Inst. Fourier, Volume 24 (1974) no. 3, pp. 171-217 | DOI | Numdam | Zbl

[15] PETER Greenberg Pseudogroups of C 1 piecewise projective homeomorphisms, Pac. J. Math., Volume 129 (1987) no. 1, pp. 67-75 | DOI | MR | Zbl

[16] Maluba Kaluba; Dawid Kielak; Piotr W. Nowak On property (T) for Aut(𝔽 n ) and SL n () (2018) (https://arxiv.org/abs/1812.03456, to appear in Annals of Mathematics)

[17] Maluba Kaluba; Piotr W. Nowak; Narutaka Ozawa Aut(𝔽 5 ) has property (T), Math. Ann., Volume 375 (2019) no. 3-4, pp. 1169-1191 | MR | Zbl

[18] Johannes Kellendonk; Mark V. Lawson Partial actions of groups, Int. J. Algebra Comput., Volume 14 (2004) no. 1, pp. 87-114 | DOI | MR | Zbl

[19] Nicolaas Hendrik Kuiper Sur les surfaces localement affines., Géométrie différentielle (Colloques Internationaux du Centre National de la Recherche Scientifique), Volume 1953, CNRS Editions, 1953, pp. 79-87 | Zbl

[20] Nicolaas Hendrik Kuiper Locally projective spaces of dimension one, Mich. Math. J., Volume 2 (1954), pp. 95-97 | MR | Zbl

[21] Yash Lodha; Nicoás Matte Bon; Michele Triestino Property FW, differentiable structures, and smoothability of singular actions, J. Topol., Volume 13 (2020) no. 3, pp. 1119-1138 | DOI | MR | Zbl

[22] Yash Lodha; Justin Tatch Mooren A nonamenable finitely presented group of piecewise projective homeomorphisms, Groups Geom. Dyn., Volume 10 (2016) no. 1, pp. 177-200 | DOI | MR | Zbl

[23] Nicolas Monod Groups of piecewise projective homeomorphisms, Proc. Natl. Acad. Sci. USA, Volume 110 (2013) no. 12, pp. 4524-4527 | DOI | MR | Zbl

[25] Bernhard Hermann Neumann Groups covered by finitely many cosets, Publ. Math., Volume 3 (1954), pp. 227-242 | MR | Zbl

[26] Yann Ollivier; Daniel T. Wise Kazhdan groups with infinite outer automorphism group, Trans. Am. Math. Soc., Volume 359 (2007) no. 5, pp. 1959-1976 | DOI | MR | Zbl

[27] Richard S. Palais A global formulation of the Lie theory of transformation groups, Memoirs of the American Mathematical Society, 22, American Mathematical Society, 1957 | MR | Zbl

[28] Bruno Poizat A course in model theory. An introduction to contemporary mathematical logic, Universitext, Springer, 2000 (translated from the French by Moses Klein) | Zbl

[29] Guyan Robertson Crofton formulae and geodesic distance in hyperbolic spaces, J. Lie Theory, Volume 8 (1998) no. 1, pp. 163-172 | MR | Zbl

[30] Guyan Robertson; T. Steger Negative definite kernels and a dynamical characterization of property (T) for countable groups, Ergodic Theory Dyn. Syst., Volume 18 (1998) no. 1, pp. 247-253 | DOI | MR | Zbl

[31] Karl Strambach Der von Staudtsche Standpunkt in lokal kompakten Geometrien, Math. Z., Volume 155 (1977) no. 1, pp. 11-21 | DOI | MR | Zbl

[32] Andrzej Żuk La propriété (T) de Kazhdan pour les groupes agissant sur les polyèdres, C. R. Math. Acad. Sci. Paris, Volume 323 (1996) no. 5, pp. 453-458 | MR | Zbl

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