Free subgroup numbers modulo prime powers: the non-periodic case

TL;DR

This paper investigates the behavior of free subgroup numbers in finitely generated virtually free groups modulo prime powers, revealing non-periodic cases where these numbers relate to binomial coefficients and enabling more efficient computations.

Contribution

It characterizes the non-periodic cases of free subgroup numbers modulo prime powers and provides formulas involving binomial coefficients for their congruences.

Findings

Non-periodic free subgroup numbers relate to binomial coefficients and rational functions.

Results enable more efficient computation of subgroup number congruences.

Provides a detailed description of subgroup number behavior in non-periodic cases.

Abstract

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, 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].