Rank Rigidity for Cat(0) Cube Complexes

By P. Caprace, Michah Sageev. 2010

220 citations recorded by the source corpus.

We prove that any group acting essentially without a fixed point at infinity on an irreducible finite-dimensional CAT(0) cube complex contains a rankone isometry. This implies that the Rank Rigidity Conjecture holds for CAT(0) cube complexes. We derive a number of other consequences for CAT(0) cube complexes, including a purely geometric proof of the Tits alternative, an existence result for regular elements in (possibly non-uniform) lattices acting on cube complexes, and a characterization of products of trees in terms of bounded cohomology.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/ae89052d6a390cdde5befdaf8db7b00b206dc55c

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Citations in the corpus

Weak hyperbolicity of cube complexes and quasi‐arboreal groups (2011, background): “This fact was proved in greater generality by Bridson [5], and a simpler proof in the cubical context appears in [7].”
Hyperbolically embedded subgroups and rotating families in groups acting on hyperbolic spaces (2011, methodology): “This conjecture was settled by Caprace and Sageev [40] for CAT (0) cube complexes.”
Acylindrically hyperbolic groups (2013, methodology): “Together with the work of Caprace– Sageev [12] and Corollary 7.3, this implies the following alternative for right angled Artin groups: every RAAG is either cyclic or directly decomposable or acylindrically hyperbolic.”
Boundaries and automorphisms of hierarchically hyperbolic spaces (2016, methodology): “13 to reprove the rank-rigidity theorem of Caprace and Sageev [CS11], in the case where the cube complex in question contains a factor system.”
GROUPS ACTING ACYLINDRICALLY ON HYPERBOLIC SPACES (2017, methodology): “Together with the work of Caprace–Sageev [17], this implies the following alternative for right angled Artin groups: every right angled Artin group is either cyclic, decomposes as a direct product of two non-trivial groups, or acylindrically hyperbolic.”
Statistically convex-cocompact actions of groups with contracting elements (2016, background): “r any n;m≥0. Moreover, the Poincare series P V ;P (s;o) diverges at s=!(V;P). Proof. Note that there exists c0 >0 such that ♯OM(i;2 ) ≤c0 exp(i⋅!) for any i≥1. Consequently, a re-arrangement of (20) gives rise to the form of (21) (see [27, Proposition 4.1] for instance). The \moreover" state”
Hyperbolicities in CAT(0) cube complexes (2017, unknown): “For fun, we mention the following consequence of Caprace and Sageev’s work [CS11].”
Contracting boundaries of CAT(0) spaces (2013, methodology): “Other notions of boundaries for CAT(0) cube complexes have been introduced by Roller [26], Guralnik [16], Nevo-Sageev [24], Hagen [17] and Behrstock-Hagen [5].”
Automorphisms of graph products of groups from a geometric perspective (2018, background): “Following [15], we say that the action of a group G on a CAT(0) cube complex Y is essential if no G-orbits stay in some neighbourhood of a half-space.”
Hierarchically hyperbolic spaces I: curve complexes for cubical groups (2014, background): “The Double-Skewering Lemma [CS11] implies that there exists g ∈ G acting hyperbolically on X whose axis skewers U and V (in the sense that U and gU are separated by V ).”
Finiteness properties of cubulated groups (2012, background): “nsion of C by [Wri]. One obtains a rich family of quasimorphisms of G from irreducible actions on a CAT(0) cube complex by recent work of FINITENESS PROPERTIES OF CUBULATED GROUPS 6 Caprace–Sageev in [CS11]. When G acts properly on a CAT(0) cube complex, then Hagen has shown that G is weakly hyperbo”
Contracting isometries of CAT(0) cube complexes and acylindrical hyperbolicity of diagram groups (2016, methodology): “Corollary 4.26 corresponds to the Double Skewering Lemma [CS11].”
The median class and superrigidity of actions on CAT(0) cube complexes (2012, methodology): “Caprace–Sageev proved [20] that there is always a decomposition of a CAT(0) cube complex analogous to the decomposition of symmetric spaces into ‘irreducible’ (or ‘rank 1’) factors.”
Growth tight actions (2014, background): “The Rank Rigidity Conjecture [18, 7] asserts that if X is a locally compact, irreducible, geodesically complete CAT(0) space, and G is an infinite discrete group acting properly and cocompactly on X , then one of the following holds: (1) X is a higher rank symmetric space.”