Search arXiv⌕ Search

arXiv · 1807.06090

Subgroup growth of all Baumslag-Solitar groups

Abstract

This paper gives asymptotic formulas for the subgroup growth and maximal subgroup growth of all Baumslag-Solitar groups.

Explore related subjects

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Andrew James Kelley. 2018-07-16. Subgroup growth of all Baumslag-Solitar groups. https://arxiv.org/abs/1807.06090

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

Some questions related to free-by-cyclic groups and tubular groups

We prove that a CAT(0) free-by-cyclic tubular group with a single vertex is virtually special, but many of them cannot virtually act freely and cocompactly on CAT(0) cube complexes. This partially confirms a question of Brady--Soroko \cite[Section 9: Question 1]{BS} and answers a question of Lyman \cite[Question 1]{Ly} in the negative. Furthermore, we provide examples of free-by-cyclic groups amalgamated along cyclic subgroups that are not virtually free-by-cyclic. This answers negatively a question of Hagen--Wise \cite[Remark 3.6]{hw}. Lastly, we exhibit an example of a cyclic-subgroup-separable tubular group that does not have the property (VRC) (i.e. every cyclic subgroup is a virtual retract). This answers a question of Minasyan \cite[Question 11.6]{min} in the negative.

math.GR↗

Soficity of One-Relator Groups and Reducible Presentations

We prove that every one-relator group is sofic, answering a question of Nate Brown. More generally, every group admitting a reducible presentation without proper powers is sofic, with one proper-power relation allowed at the final reducible step. The proof starts from the endpoint-preserving edge replacements of Poulin--Wróbel. We retain the free-group word carried by each terminal replacement and use it to define an \(F(S)\)-valued cocycle repair; in its relative form, the repair preserves the previously chosen generator coordinates while controlling the new relator defect. Coinduction transports the lower cocycle through successive one-relator-product extensions, allowing the construction to be iterated along reducible presentations. For a final relator \(w^m\), cyclic translates of the repaired set amplify a zero-defect set of measure close to \(1/m\) to one of measure close to one. A cocycle criterion then converts arbitrarily small relator defect into finite permutation approximations via a treeable skew-product orbit relation.

math.GR↗

Comparing word probabilities in finite groups

We use results from extremal graph theory and analysis to compare word probabilities in finite groups. In particular, we show that the probability of satisfying the 2-Engel word admits a positive lower bound in terms of the probability that an element has order dividing 3. More generally, using a group removal lemma, we obtain analogous bounds for $x^m$-good words.

math.GR↗