Quasi-isometric classification of graph manifold groups
By Jason A. Behrstock, Walter D. Neumann. 2006
86 citations recorded by the source corpus.
We show that the fundamental groups of any two closed irreducible nongeometric graph manifolds are quasi-isometric. We also classify the quasi-isometry types of fundamental groups of graph manifolds with boundary in terms of certain finite twocolored graphs. A corollary is the quasi-isometric classification of Artin groups whose presentation graphs are trees. In particular, any two right-angled Artin groups whose presentation graphs are trees of diameter greater than 2 are quasi-isometric; further, this quasi-isometry class does not include any other right-angled Artin groups. A finitely generated group can be considered geometrically when endowed with a word metric. Up to quasi-isometric equivalence, such metrics are unique. (In this article, only finitely generated groups are considered.) Given a collection of groups G, Gromov [11] proposed the fundamental questions of identifying which groups are quasi-isometric to those in G (rigidity) and which groups in G are quasi-isometric to each other (classification). In this article, we focus on the classification question for graph manifold groups and right-angled Artin groups. A compact 3-manifold M is called geometric if M\∂M admits a geometric structure in the sense of Thurston (i.e., a complete locally homogeneous Riemannian metric of finite volume). Thurston’s geometrization conjecture (see [29], [17], [21], [23], [22]) provides that every irreducible 3-manifold of zero Euler characteristic (i.e., with boundary consisting only of tori and Klein bottles) admits a decomposition along tori and Klein bottles into geometric pieces, the minimal such decomposition being called the geometric decomposition. There is a considerable literature on quasi-isometric rigidity and classification of 3-manifold groups. The rigidity results can be briefly summarized in the following form.
Subfields: Applied Mathematics, Geometry and Topology
View Paper: https://www.semanticscholar.org/paper/f21886147e58dcac76e768e2ae2217aa450ec196
How researchers see this paper
The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.
In Applied Mathematics
In Geometry and Topology
Citations in the corpus
- An introduction to right-angled Artin groups (2006, background): “Viewing the groups as metric spaces (via the word metric), one can ask for a weaker classification, namely which right-angled Artin groups are quasi-isometric to each other? Some partial answers to these questions have recently been found by Behrstock and Neumann [ 6 ] and by Bestvina, Kleiner, and ”
- Acylindrical hyperbolicity of groups acting on trees (2013, unknown): “We note that a classification of fundamental groups of compact 3-manifolds up to quasi-isometry was done by Behrstock andNeumann [14,15], while no non-trivial results about measure equivalence seem to be known.”