Picard modular groups generated by complex reflections

2021/12/14 by Mark, Alice, Paupert, Julien, Polletta, David
#FOS: Mathematics #Geometric Topology (math.GT) #Group Theory (math.GR)

paper · doi:10.48550/arxiv.2112.07797

Abstract

In this short note we use the presentations found in \citeMP and \citePo to show that the Picard modular groups \rm PU(2,1,O_d) with d=1,3,7 (respectively the quaternion hyperbolic lattice \rm PSp(2,1,H) with entries in the Hurwitz integer ring H) are generated by complex (resp. quaternionic) reflections, and that the Picard modular groups \rm PU(2,1,O_d) with d=2,11 have an index 4 subgroup generated by complex reflections.

Related

• Real reflections, commutators and cross-ratios in complex hyperbolic space
• On the geometry of a Picard modular group
• Generators of a Picard modular group in two complex dimensions
• Generators of Picard modular groups
• Presentations for the Euclidean Picard modular groups