Searcharxiv⌕ Search
Home/Archive/arXiv

arXiv · 1207.6548

A finitely generated branch group of exponential growth without free subgroups

Elisabeth Fink
View original source ↗Read source document ↗Export metadata ↓

Abstract

We will give an example of a branch group $G$ that has exponential growth but does not contain any non-abelian free subgroups. This answers question 16 from \cite{Bartholdi} positively. The proof demonstrates how to construct a non-trivial word $w_{a,b}(x,y)$ for any $a,b \in G$ such that $w_{a,b}(a,b) = 1$. The group $G$ is not just-infinite. We prove that every normal subgroup of $G$ is finitely generated as an abstract group and every proper quotient soluble. Further, $G$ has infinite virtual first Betti number but is not large.

Explore related subjects

math.GR

Keep this discovery

BibTeXRIS

Elisabeth Fink. 2012-07-27. A finitely generated branch group of exponential growth without free subgroups. https://arxiv.org/abs/1207.6548

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

KEEP EXPLORING

Related papers

arXiv2026

Reversibility and its asymptotic counting in Picard group

We investigate reversible elements in the Picard modular group $\mathrm{PSL}(2,\mathbb{Z}[i])$. We show that reversibility coincides with strong reversibility for Kleinian groups, in particular for the Picard group. We classify reversible elements in the Picard group and characterize loxodromic reversible elements up to conjugacy. We prove that each such conjugacy class contains exactly eight special representatives. We also obtain asymptotic estimates for the number of reversible conjugacy classes with bounded trace.

math.GR↗
arXiv2026

Conjugator length in finitely generated groups

We describe all functions $\mathbb{N}\rightarrow \mathbb{N}$ that can be realized, up to the standard equivalence, as conjugator length functions of finitely generated groups. Furthermore, we show that any two increasing functions $f,g\colon \mathbb N\to \mathbb N$ can be simultaneously realized as conjugator length functions of finitely generated, commensurable (in particular, quasi-isometric) groups.

math.GR↗
arXiv2026

The spectrum of conjugator length functions

A recent program tries to find which functions appear as conjugator length functions. In this note, we show that any (computable) increasing function larger than $n$ appears as $\mathrm{Cl}_G$ for some finitely generated (recursively presented) group. On the other hand, we demonstrate that either $\mathrm{Cl}_G$ must be constant or $\mathrm{Cl}_G(n)\succ n$. Combining these, we obtain a complete description of which functions appear as conjugator length functions of finitely generated groups.

math.GR↗