Parity patterns associated with lifts of Hecke groups

Let $q$ be an odd prime, $m$ a positive integer, and let $\Ga_m(q)$ 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, $\Ga_m(q)$ 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 $\Ga_m(q)$ which were defined in [T. W. M\"uller, Adv. in Math. 153 (2000), 118-154], and which count all the homomorphisms of index $n$ subgroups of $\Ga_m(q)$ 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 $\mathfrak H(q)=C_2*C_q$: (i) determination of the parity of the subgroup numbers of $\mathfrak H(q)$; (ii) determination of the parity of the number of index $n$ subgroups of $\mathfrak H(q)$ which are isomorphic to a free product of copies of $C_2$ and of $C_\infty$; (iii) determination of the parity of the number of index $n$ subgroups in $\mathfrak H(q)$ which are isomorphic to a free product of copies of $C_q$. The first problem has already been solved in [T. W. M\"uller, in: {\it Groups: Topological, Combinatorial and Arithmetic Aspects}, (T. W. M\"uller ed.), LMS Lecture Notes Series 311, Cambridge University Press, Cambridge, 2004, pp. 327-374]. The bulk of our paper deals with the solution of Problems (ii) and (iii).