Volume 22, issue 4 (2018)

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Hyperbolic jigsaws and families of pseudomodular groups, I

Beicheng Lou, Ser Peow Tan and Anh Duc Vo

Geometry & Topology 22 (2018) 2339–2366

We show that there are infinitely many commensurability classes of pseudomodular groups, thus answering a question raised by Long and Reid. These are Fuchsian groups whose cusp set is all of the rationals but which are not commensurable to the modular group. We do this by introducing a general construction for the fundamental domains of Fuchsian groups obtained by gluing together marked ideal triangular tiles, which we call hyperbolic jigsaw groups.

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pseudomodular, killer intervals, hyperbolic jigsaw, marked ideal triangle
Mathematical Subject Classification 2010
Primary: 11F06, 20H05, 20H15, 30F35, 30F60
Secondary: 57M05, 57M50
Received: 15 November 2016
Revised: 23 June 2017
Accepted: 9 October 2017
Published: 5 April 2018
Proposed: Ian Agol
Seconded: Anna Wienhard, Jean-Pierre Otal
Beicheng Lou
Department of Mathematics
National University of Singapore
Ser Peow Tan
Department of Mathematics
National University of Singapore
Anh Duc Vo
Department of Mathematics
National University of Singapore