The virtual Haken conjecture

By Ian Agol, D. Groves, J. Manning. 2012

618 citations recorded by the source corpus.

We prove that cubulated hyperbolic groups are virtually special. The proof relies on results of Haglund and Wise which also imply that they are linear groups, and quasi-convex subgroups are separable. A consequence is that closed hyperbolic 3-manifolds have finite-sheeted Haken covers, which resolves the virtual Haken question of Waldhausen and Thurston's virtual fibering question. An appendix to this paper by Agol, Groves, and Manning proves a generalization of the main result of "Residual finiteness, QCERF and fillings of hyperbolic groups".

View Paper: https://www.semanticscholar.org/paper/a3731b39ed8acf4928d6b698afde358fed2a2233

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Citations in the corpus

CAT(0) cube complexes and groups (2014, background): “These constructions are dual to one another: (1) Given a finite width locally finite pocset, Σ, then H(X(Σ)) ≡ Σ.”
Distinguishing geometries using finite quotients (2014, methodology): “The proof of Theorem A uses the dramatic recent developments of Agol [Ago13], Kahn–Markovic [KM12] and Wise [Wis12], whose work implies that the fundamental groups of closed hyperbolic 3-manifolds are also fundamental groups of compact, virtually special cube complexes (see [AFW12] for a summary of ”
The Thurston norm and twisted Alexander polynomials (2012, methodology): “(iii) Agol [2], building on the Surface Subgroup Theorem of Kahn–Markovic [23] and on work of Wise [49] showed that in fact the fundamental group of any closed hyperbolic 3-manifold is virtually RFRS.”
On the growth of $L^2$-invariants for sequences of lattices in Lie groups (2012, methodology): “Then we make use of a very recent result of Agol [3] to show that the fundamental group of an hypothetic compact quotient of an Nα has good separability properties (see Proposition 13.”
Absolute profinite rigidity and hyperbolic geometry (2018, background): “1 > print AbelianQuotientInvariants(l[1]); [ 6, 0 ] > l:=LowIndexSubgroups(g,<11,11>); > print #l; 0 > l:=LowIndexSubgroups(g,<12,12>); > print #l; 7 > print AbelianQuotientInvariants(l[1]); [ 0 ] > print AbelianQuotientInvariants(l[2]); [ 5, 0 ]”
Transitivity degrees of countable groups and acylindrical hyperbolicity (2015, result): “The study of graph products and their subgroups has gained additional importance in view of the recent breakthrough results of Agol, Haglund, Wise, and their co-authors, which show that many groups can be virtually embedded into right angled Artin groups (see [1, 27, 57] and references therein).”
Acylindrical hyperbolicity of groups acting on trees (2013, unknown): “claim that many groups can be virtually embedded into right angled Artin groups (see [1,41,92] and references therein).”
Nilprogressions and groups with moderate growth (2015, background): “is not solvable if n is large enough. This contradiction ends the proof of Fact 4.18. The proof of Lemma 4.21 requires the following two preliminary steps. Lemma 4.22. There is n0 ∈ Nsuch that if d ∈ [0,1], n ≥ n0, and H ≤ Sym(n) is a subgroup of order at least dnn!, then H has an orbit of size at l”