Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 1112.4832

Affine actions on non-archimedean trees

Shane O Rourke
View original source ↗Read source document ↗Export metadata ↓

Abstract

We initiate the study of affine actions of groups on $Λ$-trees for a general ordered abelian group $Λ$; these are actions by dilations rather than isometries. This gives a common generalisation of isometric action on a $Λ$-tree, and affine action on an $\R$-tree as studied by I. Liousse. The duality between based length functions and actions on $Λ$-trees is generalised to this setting. We are led to consider a new class of groups: those that admit a free affine action on a $Λ$-tree for some $Λ$. Examples of such groups are presented, including soluble Baumslag-Solitar groups and the discrete Heisenberg group.

Explore related subjects

math.GR

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Shane O Rourke. 2012-04-03. Affine actions on non-archimedean trees. https://doi.org/10.1142/s0218196713400018

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

KEEP EXPLORING

Related papers

arXiv2026

Subindices and subfactors of $\mathbb{Z}_n$ and $k$-index stability of finite groups

We study subindices, subfactors, and index stability in the cyclic group $\mathbb{Z}_n$. We prove several theorems that not only confirm a conjecture and resolve some open problems about index stability of such groups, but also provide basic tools for the characterization of finite $k$-index stable groups. As a consequence, we completely characterize all 2-element index stable subsets of $\mathbb{Z}_n$, obtain an exact closed formula for their density, and determine all $n$ for which every 2-subset is index unstable. Finally, we present some problems and a research project extending the study to 3-subsets and general $k$-subsets.

math.GR↗
arXiv2026

Virtually generating graphs of pro-$p$ groups

We study the virtually generating graph of pro-$p$ groups. We show that various classes of pro-$p$ groups have connected virtually generating graph and we bound its diameter in these cases; e.g.\ compact subgroups of analytic groups over local fields and the Nottingham group.

math.GR↗
arXiv2026

Surface subgroups of Baumslag doubles along short words

If $U$ is a minimal, diskbusting, finite list of words in a free group $F_n$ of rank $n$ such that the sum of the lengths of words in $U$ is at most $2n+4$, we prove that the natural presentation complex of the Baumslag double of $F_n$ along $U$ virtually contains a $π_1$-injective embedded closed hyperbolic surface. This verifies the Tiling Conjecture of Kim and Wilton for this type of lists of words, and in particular, implies that the corresponding Baumslag double contains a hyperbolic surface subgroup.

math.GR↗