Presentations for cusped arithmetic hyperbolic lattices

Alice Mark; Julien Paupert

doi:10.2140/agt.2022.22.3577

Presentations for cusped arithmetic hyperbolic lattices

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

Abstract

We present a general method to compute a presentation for any cusped arithmetic hyperbolic lattice Γ, applying a classical result of Macbeath to a suitable Γ–invariant horoball cover of the corresponding symmetric space. As applications we compute presentations for the Picard modular groups PU(2; 1; O_d) for d = 1; 3; 7 and the quaternion hyperbolic lattice PU(2; 1; H) with entries in the Hurwitz integer ring H. The implementation of the method for these groups is computer-assisted.

Original language	English (US)
Pages (from-to)	3577-3626
Number of pages	50
Journal	Algebraic and Geometric Topology
Volume	22
Issue number	8
DOIs	https://doi.org/10.2140/agt.2022.22.3577
State	Published - 2022
Externally published	Yes

ASJC Scopus subject areas

Geometry and Topology

Access to Document

10.2140/agt.2022.22.3577

Cite this

@article{b984a94b971b4516844422606e588022,

title = "Presentations for cusped arithmetic hyperbolic lattices",

abstract = "We present a general method to compute a presentation for any cusped arithmetic hyperbolic lattice Γ, applying a classical result of Macbeath to a suitable Γ–invariant horoball cover of the corresponding symmetric space. As applications we compute presentations for the Picard modular groups PU(2; 1; Od) for d = 1; 3; 7 and the quaternion hyperbolic lattice PU(2; 1; H) with entries in the Hurwitz integer ring H. The implementation of the method for these groups is computer-assisted.",

author = "Alice Mark and Julien Paupert",

note = "Funding Information: Paupert was partially supported by National Science Foundation grant DMS-1708463. Publisher Copyright: {\textcopyright} 2022 Mathematical Sciences Publishers.",

year = "2022",

doi = "10.2140/agt.2022.22.3577",

language = "English (US)",

volume = "22",

pages = "3577--3626",

journal = "Algebraic and Geometric Topology",

issn = "1472-2747",

publisher = "Agriculture.gr",

number = "8",

}

TY - JOUR

T1 - Presentations for cusped arithmetic hyperbolic lattices

AU - Mark, Alice

AU - Paupert, Julien

PY - 2022

Y1 - 2022

N2 - We present a general method to compute a presentation for any cusped arithmetic hyperbolic lattice Γ, applying a classical result of Macbeath to a suitable Γ–invariant horoball cover of the corresponding symmetric space. As applications we compute presentations for the Picard modular groups PU(2; 1; Od) for d = 1; 3; 7 and the quaternion hyperbolic lattice PU(2; 1; H) with entries in the Hurwitz integer ring H. The implementation of the method for these groups is computer-assisted.

AB - We present a general method to compute a presentation for any cusped arithmetic hyperbolic lattice Γ, applying a classical result of Macbeath to a suitable Γ–invariant horoball cover of the corresponding symmetric space. As applications we compute presentations for the Picard modular groups PU(2; 1; Od) for d = 1; 3; 7 and the quaternion hyperbolic lattice PU(2; 1; H) with entries in the Hurwitz integer ring H. The implementation of the method for these groups is computer-assisted.

UR - http://www.scopus.com/inward/record.url?scp=85152472395&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85152472395&partnerID=8YFLogxK

U2 - 10.2140/agt.2022.22.3577

DO - 10.2140/agt.2022.22.3577

M3 - Article

AN - SCOPUS:85152472395

SN - 1472-2747

VL - 22

SP - 3577

EP - 3626

JO - Algebraic and Geometric Topology

JF - Algebraic and Geometric Topology

IS - 8

ER -

Presentations for cusped arithmetic hyperbolic lattices

Abstract

ASJC Scopus subject areas

Access to Document

Other files and links

Fingerprint

Cite this