## FREE SUBGROUP NUMBERS MODULO PRIME POWERS: THE NON-PERIODIC CASE

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##### Journal

Journal of Combinatorial Theory, Series A

##### ISSN

0097-3165

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Show full item record##### Abstract

In [J. Algebra 452 (2016), 372–389], we characterise when the sequence
of free subgroup numbers of a finitely generated virtually free group Γ 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 λ in Γ is — essentially —
congruent to a binomial coefficient times a rational function in λ modulo a power of a
prime that divides a certain invariant of the group Γ, respectively to a binomial sum
involving such numbers. These results, apart from their intrinsic interest, in particular
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].

##### Authors

KRATTENTHALER, C; MUELLER, TW##### Collections

- College Publications [4098]