Cubulating random groups at density less than $1/6$

Ollivier, Yann; Wise, Daniel T.

doi:10.1090/s0002-9947-2011-05197-4

Cited by 76 publications

(79 citation statements)

References 21 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…In recent years much attention has been paid to groups acting properly and cocompactly on CAT(0) cube complexes, which are important examples of metric non-positive curvature and which also have features reminiscent of hyperbolicity. Such complexes give a combinatorially and geometrically rich framework to build on, and many groups have been shown to admit such actions (for a small sample, see [4,12,19,20,23]).…”

Section: Introductionmentioning

confidence: 99%

Retraction: Hierarchical hyperbolicity of all cubical groups

2017

Bulletin of London Math Soc

View full text Add to dashboard Cite

Let X be a proper CAT(0) cube complex admitting a proper cocompact action by a group G. Then X has a factor system in the sense of Behrstock, Hagen and Sisto (Preprint, 2014). By results in Behrstock, Hagen and Sisto (Preprint, 2014), it follows that G is a hierarchically hyperbolic group; this answers questions raised in Behrstock, Sisto (Preprint, 2014, 2015). Our results also resolve a conjecture of Behrstock-Hagen on boundaries of cube complexes. In particular, X cannot contain a convex staircase.

show abstract

Section: Introductionmentioning

confidence: 99%

Retraction: Hierarchical hyperbolicity of all cubical groups

2017

Bulletin of London Math Soc

View full text Add to dashboard Cite

show abstract

“…Corollary H and the results of [OW11] imply that random groups at low density do not satisfy H F D . Note however that, at density d > 1 3 , random groups are Kazhdan [Ż03, KK13], hence satisfy property H F D .…”

Section: Remarksmentioning

confidence: 96%

Superrigidity of actions on finite rank median spaces

Fioravanti¹

2019

Advances in Mathematics

View full text Add to dashboard Cite

Finite rank median spaces are a simultaneous generalisation of finite dimensional CAT(0) cube complexes and real trees. If Γ is an irreducible lattice in a product of rank-one simple Lie groups, we show that every action of Γ on a complete, finite rank median space has a global fixed point. This is in sharp contrast with the behaviour of actions on infinite rank median spaces, where even proper cocompact actions can arise.The fixed point property is obtained as corollary to a superrigidity result; the latter holds for irreducible lattices in arbitrary products of compactly generated topological groups.We exploit Roller compactifications of median spaces; these were introduced in [Fio17] and generalise a well-known construction in the case of cube complexes. We show that the Haagerup cocycle provides a reduced 1-cohomology class that detects group actions with a finite orbit in the Roller compactification. This is new even for CAT(0) cube complexes and has interesting consequences involving Shalom's property HF D . For instance, in Gromov's density model, random groups at low density do not have HF D .

show abstract

“…This applies to most finitely presented groups (where "most" means "with overwhelming probability" in one of several probabilistic models): recent results of Agol [1] and Ollivier and Wise [45] together with the older result of Olshanskii [46] imply that most finitely presented groups are linear (even over Z).…”

Section: The "Yes" and "No" Parts Of The Mckinsey Algorithmmentioning

confidence: 99%