Free subgroup numbers modulo prime powers: the non-periodic case
In [J. Algebra 452 (2016), 372-389],
we characterise when the sequence of free subgroup numbers of a
finitely generated virtually free group \Gamma
is ultimately periodic modulo a given prime power.
Here, we show that, in the remaining cases in which the sequence of
free subgroup numbers is not ultimately periodic modulo a given
prime power, the number of free subgroups of index \lambda
in \Gamma is - essentially -
congruent to a binomial coefficient times a rational function
in \lambda modulo a power of a prime that divides a certain invariant
of the group \Gamma,
respectively to a binomial sum involving such numbers.
These results allow for a much more efficient computation of
congruences for free subgroup numbers in these cases compared
to the direct recursive computation of these numbers implied
by the generating function results in [J. London Math. Soc. (2)
44 (1991), 75-94].
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