2025/05/22 by Iker de las Heras, Heras, Iker de las, Benjamin Klopsch +3
Mathematics · #Advanced Topology and Set Theory #FOS: Mathematics #Group Theory (math.GR) #Mathematical Dynamics and Fractals #advanced mathematical theories
paper · pdf · doi:10.48550/arxiv.2505.16417
openalex publication_date 2025/05/22 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We establish that finitely generated non-abelian direct products G of free pro-p groups have full Hausdorff spectrum with respect to the lower p-series L. This complements similar results with respect to other standard filtration series and a recent theorem showing that the Hausdorff spectrum hspecL(G) of a p-adic analytic pro-p group G is discrete and consists of at most 2dim(G) rational numbers. The latter also left some room for improvement regarding the upper bound. Indeed, for finitely generated nilpotent pro-p groups G we obtain the stronger assertion that the cardinality of the Hausdorff spectrum is at most the analytic dimension of G. Moreover, we produce a corresponding result when the p-adic analytic pro-p group G is just infinite, which holds not just for the lower p-series but for arbitrary filtration series. Finally, we show that, if G is a countably based pro-p group with an open subgroup mapping onto the free abelian pro-p group ℤp ⊕ ℤp, then for every prescribed finite set \0,1\ ⊆ X ⊆ [0,1] there is a filtration series S such that hspecS(G) = X; in particular, |hspecS(G)| is unbounded, as S runs through all filtration series of G with |hspecS(G)| < ∞.