Ideal bicombings for hyperbolic groups and applications

By Igor Mineyev, Nicolas Monod, Yehuda Shalom. 2003

61 citations recorded by the source corpus.

For every hyperbolic group and more general hyperbolic graphs, we construct an equivariant ideal bicombing: this is a homological analogue of the geodesic flow on negatively curved manifolds. We then construct a cohomological invariant which implies that several Measure Equivalence and Orbit Equivalence rigidity results established by Monod–Shalom hold for all non-elementary hyperbolic groups and their non-elementary subgroups. We also derive superrigidity results for actions of general irreducible lattices on a large class of hyperbolic metric spaces.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/68e4964eddd81a93386626b9e9530139a260b256

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In Geometry and Topology

1. Denis V. Osin
2. Sorin Popa
3. A. Furman
4. Nicolas Monod
5. P. Caprace
6. A. Iozzi
7. D. Groves
8. J. Manning

In Mathematical Physics

1. Denis V. Osin
2. A. Furman
3. Nicolas Monod
4. Sorin Popa
5. P. Caprace
6. A. Iozzi
7. Yehuda Shalom
8. Indira Chatterji

Citations in the corpus

• Simplicity and superrigidity of twin building lattices (2006, background): “If one assumes moreover that Isom(Y ) is non-elementary, then by [ 48, Theorem 21 ] Isom(Y ) is either virtually connected or totally disconnected.”
• Singularities, Expanders and Topology of Maps. Part 1: Homology Versus Volume in the Spaces of Cycles (2009, background): “Such Gk are constructed starting with G1, where one uses parametrized geodesic lines in Z̃ and where • depends on the hyperbolicity of a suitable parametrization (see [15] [41]).”
• Flows and joins of metric spaces (2005, background): “In [30], the following discrete–continuous dichotomy was shown for a non-elementary closed subgroup H < Isom(X) acting cocompactly on X : either”
• Orbit equivalence rigidity and bounded cohomology (2006, methodology): “In case (ii), Corollary 7.6 of [MS2] ensures 2 Creg. Case (iii) is Theorem 3 in [ MMS ].”
• Geometry of the mapping class groups I: Boundary amenability (2005, background): “This function extends to a continuous function ρ � : D →[ 0, ∞) such that ρ � (x, y) = 0 if and only if x ∩ y =∅ (see [ 29 ]).”
• A SURVEY OF MEASURED GROUP THEORY (2009, methodology): “Known examples of groups in Creg ⊂ C include groups admitting ”hyperbolic-like” actions of the following types: (see [87], [83])”
• Induced quasicocycles on groups with hyperbolically embedded subgroups (2012, background): “Recall that the class Creg of Monod-Shalom is the class of groups for which H2b (G, `2(G)) 6= 0.”
• Dehn filling in relatively hyperbolic groups (2006, background): “Applications of Mineyev’s construction are myriad (see, for example, [22, 23, 24, 25, 26, 33]).”
• Isometry groups of non‐positively curved spaces: discrete subgroups (2009, background): “This is well-known; for a proof, see for example Lemma 22 in [62].”
• Strong rigidity of II1 factors arising from malleable actions of w-rigid groups, II (2003, background): “× ...× Γn, with n≥ 2, Γi torsion free, ICC and in the class C of Monod-Shalom ∀i(for instance Γi a free product of torsion free groups, cf. [MoSh], or Γi a non-elementary Gromov-hyperbolic group, cf. [MiMoSh]), at least one of which being required to have the property (T). Note that there are contin”
• The median class and superrigidity of actions on CAT(0) cube complexes (2012, methodology): “The same philosophy has been promoted by Monod [39], Monod–Shalom [40, 41] and Mineyev–Monod–Shalom [37].”