Weak hyperbolicity of cube complexes and quasi‐arboreal groups

By Mark F. Hagen. 2011

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We examine a graph Γ encoding the intersection of hyperplane carriers in a CAT(0) cube complex X~ . The main result is that Γ is quasi‐isometric to a tree. This implies that a group G acting properly and cocompactly on X~ is weakly hyperbolic relative to the hyperplane stabilizers. Using Wright's recent result on the asymptotic dimension of CAT(0) cube complexes, we give a generalization of a theorem of Bell and Dranishnikov on the finite asymptotic dimension of graphs of asymptotically finite‐dimensional groups. Finally, we apply contact graph techniques to prove a cubical version of the flat plane theorem stated in terms of complete bipartite subgraphs of Γ.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/a1791388491433fc6429749d0f3bc48502ec582a

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In Geometry and Topology

In Mathematical Physics

Citations in the corpus

Finiteness properties of cubulated groups (2012, background): “SS PROPERTIES OF CUBULATED GROUPS 6 of Caprace–Sageev in [CS11]. When G acts properly on a CAT(0) cube complex, then Hagen has shown that G is weakly hyperbolic relative to the hyperplane stabilizers [Hag]. 1.3. Survey of known results towards cubulation of groups. There has been an increasing amoun”
Hierarchically hyperbolic spaces I: curve complexes for cubical groups (2014, background): “rallelism. Throughout, Xis a CAT(0) cube complex and His the set of hyperplanes. Unless stated otherwise,weworkinthe1–skeletonofX,andwedenotebyd XthegraphmetriconX(1). The contact graph CX, deﬁned in [Hag14], is the graph whose vertex set is H, with two verticesadjacentifthecorrespondinghyperplanesa”
Hierarchically hyperbolic spaces II: Combination theorems and the distance formula (2015, background): “lly,alocalcoarsestructureofa cubecomplexasmadeprecisein[Bow13],applicationstoembeddingsintrees,[BDS11],etc. Accordingly,itisnaturaltoseekacommonthreadjoiningtheseimportantclassesofgroups andspaces. In[Hag14]itwasshownthat,foranarbitraryCAT(0)cubecomplexX,theintersectiongraph of the hyperplane carrie”
Hyperbolicities in CAT(0) cube complexes (2017, background): “) Instead, Hagen introduced in [Hag14] the contact graph ΓX of X as the graph whose vertices are the hyperplanes of X and whose edges link two hyperplanes whose carriers intersect.”
Boundaries and automorphisms of hierarchically hyperbolic spaces (2016, background): “there exists ξě 0 such that for all F,F1 P F, either gFpF1q P For diampgFpF1qq ď ξ, (4) there exists ∆ ě 1 such that each point in X belongs to at most ∆ elements of F. The contact graph CX of X (see [Hag14]) has a vertex for each hyperplane, with two hyperplanes joined by an edge if no third hyperp”
Asymptotic dimension and small‐cancellation for hierarchically hyperbolic spaces and groups (2015, background): “This is a coarse version of what happens, for example, when X is a CAT(0) cube complex with a factor system and we can take CS to be the contact graph of X (see [1, 21]).”