Special Cube Complexes

By Frédéric Haglund, Daniel T. Wise. The Structure of Groups with a Quasiconvex Hierarchy, 2008

540 citations recorded by the source corpus.

Subfields: Discrete Mathematics and Combinatorics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/679a5529a66a20859510f798cb3041df9929c629

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Citations in the corpus

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).”
Weak hyperbolicity of cube complexes and quasi‐arboreal groups (2011, background): “by the 1-skeleton and each parallelism-class of 1-cubes (explained in, for example, [12, 22]) is already represented in X̃ and thus in X̃ ′.”
Tits alternatives for graph products (2011, background): “The family of subgroups of right angled Artin groups is very rich and includes many examples (see [4, 16, 15, 9, 25]).”
Hereditary conjugacy separability of right angled Artin groups and its applications (2009, background): “Wise introduced a new class of special (or A-special, in the terminology of [32]) cube complexes, that admit a combinatorial local isometry to the Salvetti cube complex (see [11]) of a right angled Artin group (possibly, infinitely generated).”
Finiteness properties of cubulated groups (2012, methodology): “Exploiting their combinatorial geometry, a connection has been developed by Haglund–Wise in [HW08], which has the potential to virtually embed a nicely cubulated group G into a right-angled Artin group, and thus expose many algebraic features of G.”
Right-angled Artin groups and a generalized isomorphism problem for finitely generated subgroups of mapping class groups (2010, background): “the recent work of Haglund and Wise in [30]).”
Criteria for virtual fibering (2007, methodology): “We also thank Misha Kapovich for helpful comments and we thank Haglund and Wise for sharing their preprint [15].”
Property A and CAT(0) cube complexes (2009, background): “These spaces, which are generalizations of (products of simplicial) trees, appear naturally in many problems in geometric group theory and low dimensional topology [AR90, HW07 , Sag97, Wis04].”
The Thurston norm and twisted Alexander polynomials (2012, background): “The only thing we need is the following theorem of Haglund and Wise [22] and Agol [1, Theorem 2.”
RESEARCH ANNOUNCEMENT: THE STRUCTURE OF GROUPS WITH A QUASICONVEX HIERARCHY (2009, methodology): “We refer the reader to [21] and especially [20] where a version of the following criterion is given that works in the presence of torsion.”
Finite index subgroups of graph products (2008, methodology): “Remark Since the first version of this paper was written we have developed with Dani Wise a theory of groups acting specially on CAT(0) cube complexes (see [22]).”
Quasiflats in hierarchically hyperbolic spaces (2017, background): “a CATp0q{isometrically embedded copy of r0;8q with the Euclidean metric. It was established in [BHS17b] that all CAT(0) cube complexes with proper, cocompact, cospecial (in the sense of Haglund-Wise [HW08]) group actions admit factor systems. More generally, it is shown in [HS18] that a CAT(0) cube ”
Mixed 3-manifolds are virtually special (2012, background): “of R(X) labeled by the generator corresponding to A. Note that R(X) is compact special. Our goal is to revisit and strengthen hyperplane separability and double hyperplane separability established in [HW08] for compact special cube complexes. The starting point and the main tool is the following. MI”
A metric Kan–Thurston theorem (2010, background): “Special cube complexes, as introduced by Haglund and Wise in [14] are a class of cube complexes whose fundamental groups are residually finite.”
Acylindrical hyperbolicity of groups acting on trees (2013, unknown): “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 claim that many groups can be virtually embedded into right angled Artin groups (see [1,41,92] and references therein).”
Hierarchically hyperbolic spaces I: curve complexes for cubical groups (2014, background): “hF F ratherthantHXF: HPFu.) 8.2. Examplesoffactorsystems. HIERARCHICALLY HYPERBOLIC SPACES I: CURVE COMPLEXES FOR CUBICAL GROUPS 19 8.2.1. Specialgroups. Universalcoversofspecialcubecomplexes,deﬁnedin[HW08],contain factorsystemsprovidedtheco-specialactionhasﬁnitelymanyorbitsofhyperplanes,aswe shalls”
The Geometry and Topology of Coxeter Groups (2025, background): “These nice cube complexes are the “special” cube complexes of [12].”
The virtual Haken conjecture (2012, background): “We expect the reader to be familiar with non-positively curved (NPC) cube complexes [8], special cube complexes [21], and hyperbolic groups [16].”