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Spectral rigidity and discreteness of 2233-groups

Published online by Cambridge University Press:  01 January 2008

PETER BUSER ,
NICOLE FLACH ,
KLAUS-DIETER SEMMLER ,
COLIN MACLACHLAN  and
GERHARD ROSENBERGER
PETER BUSER
Affiliation:
Section de Mathématiques, École Polytechnique Fédérale de Lausanne, Station 8, CH-1015 Lausanne, Switzerland. e-mail: peter.buser@epfl.ch, nicole.flach@bluewin.ch, klaus-dieter.semmler@epfl.ch
NICOLE FLACH
Affiliation:
Section de Mathématiques, École Polytechnique Fédérale de Lausanne, Station 8, CH-1015 Lausanne, Switzerland. e-mail: peter.buser@epfl.ch, nicole.flach@bluewin.ch, klaus-dieter.semmler@epfl.ch
KLAUS-DIETER SEMMLER
Affiliation:
Section de Mathématiques, École Polytechnique Fédérale de Lausanne, Station 8, CH-1015 Lausanne, Switzerland. e-mail: peter.buser@epfl.ch, nicole.flach@bluewin.ch, klaus-dieter.semmler@epfl.ch
COLIN MACLACHLAN
Affiliation:
Department of Mathematical Sciences, University of Aberdeen, Aberdeen AB24 3UE, Scotland. e-mail: C.Maclachlan@maths.abdn.ac.uk
GERHARD ROSENBERGER
Affiliation:
Fachbereich Mathematik, Lehrstuhl LSVI, Universität Dortmund, Vogelpothsweg 87, D-44227 Dortmund, Germany. e-mail: Gerhard.Rosenberger@math.uni-dortmund.de
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Abstract

In this paper we describe methods for dealing with the trace spectrum of a subgroup of PSL(2, ) generated by four elliptic elements α, β, γ, δ of respective orders 2, 2, 3, 3, satisfying αβγδ = 1. We give a parametrization and a fundamental domain in the parameter space of such groups. Furthermore we construct an algorithm that decides whether or not a given group is discrete and which moves the discrete groups into the fundamental domain. Our main result is that any two discrete such groups are isospectral if and only if they are conjugate in (2, ).

In the Appendix we consider pairs of subgroups of (2, ) that arise from non-conjugate maximal orders in a quaternion algebra over a number field. We show that for the isospectrality of such pairs there is a peculiar exception in the case where the groups contain elements of both orders 2 and 3.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 144 , Issue 1 , January 2008 , pp. 145 - 180
Copyright
Copyright © Cambridge Philosophical Society 2008

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