ON COARSE EMBEDDINGS OF AMENABLE GROUPS INTO HYPERBOLIC GRAPHS

Romain Tessera

doi:10.1090/proc/17386

ON COARSE EMBEDDINGS OF AMENABLE GROUPS INTO HYPERBOLIC GRAPHS

Romain Tessera

Université Paris Cité

Résultats de recherche: Contribution à un journal › Article › Revue par des pairs

Résumé

In this note we prove that if a finitely generated amenable group admits a regular map to Hⁿ ×R^d, then it must be virtually nilpotent of degree of growth at most d + n − 1. This is sharp as Zⁿ+^d−¹ coarsely embeds into Hⁿ × R^d. We deduce that an amenable group regularly (or coarsely) embeds into a hyperbolic group if and only if it is virtually nilpotent, answering a question of Hume and Sisto [New York J. Math. 23 (2017), pp. 1657–1670]. We describe an application to Lorentz geometry due to Charles Frances [Geom. Funct. Anal. 31 (2021), pp. 1095–1159].

langue originale	Anglais
Pages (de - à)	4545-4552
Nombre de pages	8
journal	Proceedings of the American Mathematical Society
Volume	153
Numéro de publication	11
Les DOIs	https://doi.org/10.1090/proc/17386
état	Publié - 1 nov. 2025
Modification externe	Oui

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title = "ON COARSE EMBEDDINGS OF AMENABLE GROUPS INTO HYPERBOLIC GRAPHS",

abstract = "In this note we prove that if a finitely generated amenable group admits a regular map to Hn ×Rd, then it must be virtually nilpotent of degree of growth at most d + n − 1. This is sharp as Zn+d−1 coarsely embeds into Hn × Rd. We deduce that an amenable group regularly (or coarsely) embeds into a hyperbolic group if and only if it is virtually nilpotent, answering a question of Hume and Sisto [New York J. Math. 23 (2017), pp. 1657–1670]. We describe an application to Lorentz geometry due to Charles Frances [Geom. Funct. Anal. 31 (2021), pp. 1095–1159].",

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N2 - In this note we prove that if a finitely generated amenable group admits a regular map to Hn ×Rd, then it must be virtually nilpotent of degree of growth at most d + n − 1. This is sharp as Zn+d−1 coarsely embeds into Hn × Rd. We deduce that an amenable group regularly (or coarsely) embeds into a hyperbolic group if and only if it is virtually nilpotent, answering a question of Hume and Sisto [New York J. Math. 23 (2017), pp. 1657–1670]. We describe an application to Lorentz geometry due to Charles Frances [Geom. Funct. Anal. 31 (2021), pp. 1095–1159].

AB - In this note we prove that if a finitely generated amenable group admits a regular map to Hn ×Rd, then it must be virtually nilpotent of degree of growth at most d + n − 1. This is sharp as Zn+d−1 coarsely embeds into Hn × Rd. We deduce that an amenable group regularly (or coarsely) embeds into a hyperbolic group if and only if it is virtually nilpotent, answering a question of Hume and Sisto [New York J. Math. 23 (2017), pp. 1657–1670]. We describe an application to Lorentz geometry due to Charles Frances [Geom. Funct. Anal. 31 (2021), pp. 1095–1159].

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ER -

ON COARSE EMBEDDINGS OF AMENABLE GROUPS INTO HYPERBOLIC GRAPHS

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