Presentations for cusped arithmetic hyperbolic lattices

Mark, Alice; Paupert, Julien

doi:10.2140/agt.2022.22.3577

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Presentations for cusped arithmetic hyperbolic lattices

Alice Mark and Julien Paupert

Algebraic & Geometric Topology 22 (2022) 3577–3626

DOI: 10.2140/agt.2022.22.3577

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.

Keywords

arithmetic lattices, group presentations, hyperbolic geometry

Mathematical Subject Classification 2010

Primary: 20F05, 20F65, 22E40

References

Publication

Received: 27 February 2018

Revised: 24 June 2021

Accepted: 20 July 2021

Published: 14 March 2023

Authors

Alice Mark
Department of Mathematics Vanderbilt University Nashville, TN United States

Julien Paupert
School of Mathematical and Statistical Sciences Arizona State University Tempe, AZ United States

Volume 22, issue 8 (2022)