Parity patterns associated with lifts of Hecke gro

Authors: C Krattenthaler Thomas W Müller

Publish Date: 2008/10/22

Volume: 78, Issue: 1, Pages: 99-147

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Abstract

Let q be an odd prime m a positive integer and let Γ m q be the group generated by two elements x and y subject to the relations x 2m =y qm =1 and x 2=y q that is Γ m q is the free product of two cyclic groups of orders 2m respectively qm amalgamated along their subgroups of order m Our main result determines the parity behaviour of the generalized subgroup numbers of Γ m q which were defined in Müller Adv Math 153118–154 2000 and which count all the homomorphisms of index n subgroups of Γ m q into a given finite group H in the case when gcd m H =1 This computation depends upon the solution of three counting problems in the Hecke group ℋq=C 2C q i determination of the parity of the subgroup numbers of ℋq ii determination of the parity of the number of index n subgroups of ℋq which are isomorphic to a free product of copies of C 2 and of C ∞ iii determination of the parity of the number of index n subgroups in ℋq which are isomorphic to a free product of copies of C q The first problem has already been solved in Müller Groups Topological Combinatorial and Arithmetic Aspects LMS Lecture Notes Series vol 311 pp 327–374 Cambridge University Press Cambridge 2004 The bulk of our paper deals with the solution of Problems ii and iii

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