Finite-dimensional CAT Research Articles

Homotopy equivalent boundaries of cube complexes

A finite-dimensional CAT(0) cube complex X is equipped with several well-studied boundaries. These include the Tits boundary∂TX\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\partial _TX$$\\end{document} (which depends on the CAT(0) metric), the Roller boundary∂RX\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\partial _R}X$$\\end{document} (which depends only on the combinatorial structure), and the simplicial boundary∂▵X\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\partial _\ riangle X$$\\end{document} (which also depends only on the combinatorial structure). We use a partial order on a certain quotient of ∂RX\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\partial _R}X$$\\end{document} to define a simplicial Roller boundary R▵X\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathfrak {R}}_\ riangle X$$\\end{document}. Then, we show that ∂TX\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\partial _TX$$\\end{document}, ∂▵X\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\partial _\ riangle X$$\\end{document}, and R▵X\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathfrak {R}}_\ riangle X$$\\end{document} are all homotopy equivalent, Aut(X)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ ext {Aut}(X)$$\\end{document}-equivariantly up to homotopy. As an application, we deduce that the perturbations of the CAT(0) metric introduced by Qing do not affect the equivariant homotopy type of the Tits boundary. Along the way, we develop a self-contained exposition providing a dictionary among different perspectives on cube complexes.

Semi-simple actions of the Higman-Thompson groups T_n on finite-dimensional CAT(0) spaces

In this paper, we study isometric actions on finite-dimensional CAT(0) spaces for the Higman–Thompson groups T_n, which are generalizations of Thompson’s group T. It is known that every semi-simple action of T on a complete CAT(0) space of finite covering dimension has a global fixed point. After this result, we show that every semi-simple action of T_n on a complete CAT(0) space of finite covering dimension has a global fixed point. In the proof, we regard T_n as ring groups of homeomorphisms of S^1 introduced by Kim, Koberda and Lodha, and use general facts on these groups.

Projection complexes and quasimedian maps

We use the projection complex machinery of Bestvina--Bromberg--Fujiwara to study hierarchically hyperbolic groups. In particular, we show that if the group has a BBF colouring and its associated hyperbolic spaces are quasiisometric to trees, then the group is quasiisometric to a finite-dimensional CAT(0) cube complex. We deduce various properties, including the Helly property for hierarchically quasiconvex subsets.

The 𝑀_{𝑑}-approximation property and unitarisability

We define a strengthening of the Haagerup–Kraus approximation property by means of the subalgebras of Herz–Schur multipliers M d ( G ) M_d(G) ( d ≥ 2 d\geq 2 ) introduced by Pisier. We show that unitarisable groups satisfying this property for all d ≥ 2 d\geq 2 are amenable. Moreover, we show that groups acting properly on finite-dimensional CAT(0) cube complexes satisfy M d M_d -AP for all d ≥ 2 d\geq 2 . We also give examples of non-weakly amenable groups satisfying M d M_d -AP for all d ≥ 2 d\geq 2 .

Cubulation of some triangle-free Artin groups

We prove that some classes of triangle-free Artin groups act properly on locally finite, finite-dimensional CAT(0) cube complexes. In particular, this provides the first examples of Artin groups that are properly cubulated but cannot be cocompactly cubulated, even virtually. The existence of such a proper action has many interesting consequences for the group, notably the Haagerup property, and the Baum–Connes conjecture with coefficients.

Shortcut graphs and groups

We introduce shortcut graphs and groups. Shortcut graphs are graphs in which cycles cannot embed without metric distortion. Shortcut groups are groups which act properly and cocompactly on shortcut graphs. These notions unify a surprisingly broad family of graphs and groups of interest in geometric group theory and metric graph theory, including: the 1-skeletons of systolic and quadric complexes (in particular finitely presented C(6) and C(4)-T(4) small cancellation groups), 1-skeletons of finite dimensional CAT(0) cube complexes, hyperbolic graphs, standard Cayley graphs of finitely generated Coxeter groups and the standard Cayley graph of the Baumslag-Solitar group BS(1,2). Most of these examples satisfy a strong form of the shortcut property. The shortcut properties also have important geometric group theoretic consequences. We show that shortcut groups are finitely presented and have exponential isoperimetric and isodiametric functions. We show that groups satisfying the strong form of the shortcut property have polynomial isoperimetric and isodiametric functions.

A curiously cubulated group

We construct a finitely generated 2-dimensional group that acts properly on a locally finite CAT(0) cube complex but does not act properly on a finite dimensional CAT(0) cube complex.

On the Baum–Connes Conjecture for Groups Acting on CAT(0)-Cubical Spaces

Abstract We give a new proof of the Baum–Connes conjecture with coefficients for any second countable, locally compact topological group that acts properly and cocompactly on a finite-dimensional CAT(0)-cubical space with bounded geometry. The proof uses the Julg–Valette complex of a CAT(0)-cubical space introduced by the 1st three authors and the direct splitting method in Kasparov theory developed by the last author.

A note on torsion subgroups of groups acting on finite-dimensional CAT(0) cube complexes

In this article, we state and prove a general criterion which prevents some groups from acting properly on finite-dimensional CAT(0) cube complexes. As an application, we show that, for every non-trivial finite group F, the lamplighter group F≀F2 over a free group does not act properly on a finite-dimensional CAT(0) cube complex (although it acts properly on an infinite-dimensional CAT(0) cube complex). We also deduce from this general criterion that, given a group G acting on a CAT(0) cube complex of finite dimension and an infinite torsion subgroup L≤G, either the normaliser NG(L) is virtually (locally finite)-by-(free abelian) or, for every k≥1, NG(L) contains a non-abelian free subgroup commuting with a subgroup of L of size ≥k.

A note on locally elliptic actions on cube complexes

We deduce from Sageev's results that whenever a group acts locally elliptically on a finite dimensional CAT(0) cube complex, then it must fix a point. As an application, we give an example of a group G such that G does not have property (T), but G and all its finitely generated subgroups can not act without a fixed point on a finite dimensional CAT(0) cube complex, answering a question by Barnhill and Chatterji.

Superrigidity of actions on finite rank median spaces

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 [48] 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 HFD. For instance, in Gromov's density model, random groups at low density do not have HFD.

On groups whose actions on finite-dimensional CAT(0) spaces have global fixed points

Abstract We give a criterion for group elements to have fixed points with respect to a semi-simple action on a complete CAT(0) space of finite topological dimension. As an application, we show that Thompson’s group T and various generalizations of Thompson’s group V have global fixed points when they act semi-simply on finite-dimensional complete CAT(0) spaces, while it is known that T and V act properly on infinite-dimensional CAT(0) cube complexes.

Direct splitting method for the Baum–Connes conjecture

We develop a new method for studying the Baum–Connes conjecture, which we call the direct splitting method. We introduce what we call property (γ) for G-equivariant Kasparov cycles. We show that the existence of a G-equivariant Kasparov cycle with property (γ) implies the split-injectivity of the assembly map μAG for any separable G-C⁎-algebraA. We also show that if such a cycle exists, the assembly map μAG is an isomorphism if and only if the cycle acts as the identity on the right-hand side group K⁎(A⋊rG) of the Baum–Connes conjecture. In a separate paper, with J. Brodzki, E. Guentner and N. Higson, we use this method to give a finite-dimensional proof of the Baum–Connes conjecture for groups which act properly and co-compactly on a finite-dimensional CAT(0)-cubical space.

A differential complex for CAT(0) cubical spaces

In the 1980's Pierre Julg and Alain Valette, and also Tadeusz Pytlik and Ryszard Szwarc, constructed and studied a certain Fredholm operator associated to a simplicial tree. The operator can be defined in at least two ways: from a combinatorial flow on the tree, similar to the flows in Forman's discrete Morse theory, or from the theory of unitary operator-valued cocycles. There are applications of the theory surrounding the operator to C⁎-algebra K-theory, to the theory of completely bounded representations of groups that act on trees, and to the Selberg principle in the representation theory of p-adic groups. The main aim of this paper is to extend the constructions of Julg and Valette, and Pytlik and Szwarc, to CAT(0) cubical spaces. A secondary aim is to illustrate the utility of the extended construction by developing an application to operator K-theory and giving a new proof of K-amenability for groups that act properly on finite dimensional CAT(0)-cubical spaces.

Radial Schur multipliers on some generalisations of trees

We give a characterisation of radial Schur multipliers on finite products of trees. The equivalent condition is that a certain generalised Hankel matrix involving the discrete derivatives of the radial function is a trace class operator. This extends Haagerup, Steenstrup and Szwarc's result for trees. The same condition can be expressed in terms of Besov spaces on the torus. We also prove a similar result for products of hyperbolic graphs and provide a sufficient condition for a function to define a radial Schur multiplier on a finite dimensional CAT(0) cube complex.

Hyperbolic and cubical rigidities of Thompson’s group V

Abstract In this article, we state and prove a general criterion allowing us to show that some groups are hyperbolically elementary, meaning that every isometric action of one of these groups on a Gromov-hyperbolic space either fixes a point at infinity, or stabilises a pair of points at infinity, or has bounded orbits. Also, we show how such a hyperbolic rigidity leads to fixed-point properties on finite-dimensional CAT(0) cube complexes. As an application, we prove that Thompson’s group V is hyperbolically elementary, and we deduce that it satisfies Property ( FW ∞ ) {({\rm FW}_{\infty})} , i.e., every isometric action of V on a finite-dimensional CAT(0) cube complex fixes a point. It provides the first example of a (finitely presented) group acting properly on an infinite-dimensional CAT(0) cube complex such that all its actions on finite-dimensional CAT(0) cube complexes have global fixed points.

Classifying virtually special tubular groups

A group is tubular if it acts on a tree with \mathbb{Z}^2 vertex stabilizers and \mathbb{Z} edge stabilizers. We prove that a tubular group is virtually special if and only if it acts freely on a locally finite CAT(0) cube complex. Furthermore, we prove that if a tubular group acts freely on a finite dimensional CAT(0) cube complex, then it virtually acts freely on a three dimensional CAT(0) cube complex.

A generalized axis theorem for cube complexes

We consider a finitely generated virtually abelian group $G$ acting properly and without inversions on a CAT(0) cube complex $X$. We prove that $G$ stabilizes a finite dimensional CAT(0) subcomplex $Y \subseteq X$ that is isometrically embedded in the combinatorial metric. Moreover, we show that $Y$ is a product of finitely many quasilines. The result represents a higher dimensional generalization of Haglund's axis theorem.

The Furstenberg–Poisson boundary and CAT(0) cube complexes

We show under weak hypotheses that $\unicode[STIX]{x2202}X$, the Roller boundary of a finite-dimensional CAT(0) cube complex $X$ is the Furstenberg–Poisson boundary of a sufficiently nice random walk on an acting group $\unicode[STIX]{x1D6E4}$. In particular, we show that if $\unicode[STIX]{x1D6E4}$ admits a non-elementary proper action on $X$, and $\unicode[STIX]{x1D707}$ is a generating probability measure of finite entropy and finite first logarithmic moment, then there is a $\unicode[STIX]{x1D707}$-stationary measure on $\unicode[STIX]{x2202}X$ making it the Furstenberg–Poisson boundary for the $\unicode[STIX]{x1D707}$-random walk on $\unicode[STIX]{x1D6E4}$. We also show that the support is contained in the closure of the regular points. Regular points exhibit strong contracting properties.

Asymptotic geometry in higher products of rank one Hadamard spaces

Given a product X of locally compact rank one Hadamard spaces, we study asymptotic properties of certain discrete isometry groups of X. First we give a detailed description of the structure of the geometric limit set and relate it to the limit cone; moreover, we show that the action of on a quotient of th e regular geometric boundary of X is minimal and proximal. This is completely analogous to the case of Zariski dense discrete subgroups of semi-simple Lie groups acting on the associated symmetric space (compare [Ben97]). In the second part of the paper we study the distribution of -orbit points in X: As a generalization of the critical exponent δ() of we consider for any θ ∈ R r0 , kθk = 1, the exponential growth rate δ�() of the number of orbit points in X with prescribed “slope” θ. In analogy to Quint’s result in [Qui02b] we show that the homogeneous extension � to R r0 of δ�() as a function of θ is upper semi-continuous, concave and strictly positive in the relative interior of the limit cone. This shows in particular that there exists a unique slope θ � for which δ��() is maximal and equal to the critical exponent of . We notice that an interesting class of product spaces as above comes from the second alternative in the Rank Rigidity Theorem ( [CS11, Theorem A]) for CAT(0)-cube complexes: Given a finite-dimensional CAT(0)-cube complex X and a group of automorphisms without fixed point in the geometric comp actification of X, then either contains a rank one isometry or there exists a conve x -invariant subcomplex of X which is a product of two unbounded cube subcomplexes; in the latter case one inductively gets a convex -invariant s ubcomplex of X which can be decomposed into a finite product of rank one Hadamard spaces. So our results imply in particular that classical properties of discrete subgroups of higher rank Lie groups as in [Ben97] and [Qui02b] also hold for certain discrete isometry groups of reducible CAT(0)-cube complexes.