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Presentations for cusped arithmetic hyperbolic lattices

Alice Mark and Julien Paupert

Algebraic & Geometric Topology 22 (2022) 3577–3626
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,) with entries in the Hurwitz integer ring . 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