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

Research output: Contribution to journal › Article › peer-review

Abstract

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].

Original language	English
Pages (from-to)	4545-4552
Number of pages	8
Journal	Proceedings of the American Mathematical Society
Volume	153
Issue number	11
DOIs	https://doi.org/10.1090/proc/17386
Publication status	Published - 1 Nov 2025
Externally published	Yes

Access to Document

10.1090/proc/17386

Cite this

@article{dbbe19960c094d9eb883e9ce1723c209,

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].",

author = "Romain Tessera",

note = "Publisher Copyright: {\textcopyright} 2025 American Mathematical Society",

year = "2025",

month = nov,

day = "1",

doi = "10.1090/proc/17386",

language = "English",

volume = "153",

pages = "4545--4552",

journal = "Proceedings of the American Mathematical Society",

issn = "0002-9939",

number = "11",

}

TY - JOUR

T1 - ON COARSE EMBEDDINGS OF AMENABLE GROUPS INTO HYPERBOLIC GRAPHS

AU - Tessera, Romain

PY - 2025/11/1

Y1 - 2025/11/1

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].

UR - https://www.scopus.com/pages/publications/105020763606

U2 - 10.1090/proc/17386

DO - 10.1090/proc/17386

M3 - Article

AN - SCOPUS:105020763606

SN - 0002-9939

VL - 153

SP - 4545

EP - 4552

JO - Proceedings of the American Mathematical Society

JF - Proceedings of the American Mathematical Society

IS - 11

ER -

ON COARSE EMBEDDINGS OF AMENABLE GROUPS INTO HYPERBOLIC GRAPHS

Abstract

Access to Document

Other files and links

Fingerprint

Cite this