Motivated by the observation that groups can be effectively studied using metric spaces modelled on l1, l2, and l∞ geometry, we consider cell complexes equipped with an lp metric for arbitrary p. Under weak conditions that can be checked locally, we establish nonpositive curvature properties of these complexes, such as Busemann- convexity and strong bolicity. We also provide detailed information on the geodesics of these metrics in the special case of CAT(0) cube complexes.
Consider a closed surface M with negative Euler characteristic, and an admissible probability measure on the fundamental group of M with finite first moment. Corresponding to each point in the Teichmüller space of M, there is an associated random walk on the hyperbolic plane. We show that the speed of this random walk is a proper function on the Teichmüller space of M, and we relate the growth of the speed to the Teichmüller distance to a basepoint. One key argument is an adaptation of Gou\"ezel's pivoting techniques to actions of a fixed group on a sequence of hyperbolic metric spaces.
We motivate the study of metric spaces with a unique convex geodesic bicombing, which we call CUB spaces. These encompass many classical notions of nonpositive curvature, such as CAT(0) spaces and Busemann-convex spaces. Groups having a geometric action on a CUB space enjoy numerous properties.
We want to know when a simplicial complex, endowed with a natural polyhedral metric, is CUB. We establish a link condition, stating essentially that the complex is locally a lattice. This generalizes Gromov's link condition for cube complexes, for the ℓ∞ metric.
The link condition applies to numerous examples, including Euclidean buildings, simplices of groups, Artin complexes of Euclidean Artin groups, (weak) Garside groups, some arcs and curve complexes, and minimal spanning surfaces of knots.
In this article we study combinatorial non-positive curvature aspects of various simplicial complexes with natural An tilde shaped simplicies, including Euclidean buildings of type An tilde and Cayley graphs of Garside groups and their quotients by the Garside elements.
All these examples fit into the more general setting of lattices with order-increasing ℤ-actions and the associated lattice quotients proposed in a previous work by the first named author. We show that both the lattice quotients and the lattices themselves give rise to weakly modular graphs, which is a form of combinatorial non-positive curvature.
We also show that several other complexes fit into this setting of lattices/lattice quotients, hence our result applies, including Artin complexes of Artin-Tits groups of type An tilde, a class of arc complexes and weak Garside groups arising from a categorical Garside structure in the sense of Bessis. Along the way, we also clarify the relationship between categorical Garside structure, lattices with ℤ action and different classes of complexes studied this article.
Extending and unifying a number of well-known conjectures and open questions, we conjecture that locally elliptic (that is, every element has a bounded orbit) actions by automorphisms of finitely generated groups on finite dimensional nonpositively curved complexes have global fixed points. In particular, finitely generated torsion groups cannot act without fixed points on such spaces.
We prove these conjectures for a wide class of complexes, including all infinite families of Euclidean buildings, Helly complexes, some graphical small cancellation and systolic complexes, uniformly locally finite Gromov hyperbolic graphs. We present numerous consequences of these result, e.g. concerning the automatic continuity.
On the way we prove several results concerning automorphisms of Helly graphs. They are of independent interest and include a classification result: any automorphism of a Helly graph with finite combinatorial dimension is either elliptic or hyperbolic, with rational translation length. One consequence is that groups with distorted elements cannot act properly on such graphs. We also present and study a new notion of geodesic clique paths. Their local-to-global properties are crucial in our proof of ellipticity results.
Starting with a lattice with an action of Z or R, we build a Helly graph or an injective metric space. We deduce that the sup orthoscheme complex of any bounded graded lattice is injective. We also prove a Cartan-Hadamard result for locally injective metric spaces.
We apply this to show that any Garside group acts on an injective metric space and on a Helly graph. We also deduce that the natural piecewise sup metric on any Euclidean building of type A extended, B, C or D is injective, and its thickening is a Helly graph.
Concerning Artin groups of Euclidean types A and C, we show that the natural piecewise sup metric on the Deligne complex is injective, the thickening is a Helly graph, and it admits a convex bicombing. This gives a metric proof of the K(π,1) conjecture, as well as several other consequences usually known when the Deligne complex has a CAT(0) metric.
We define the notion of a negatively curved tangent bundle of a metric measured space. We prove that, when a group G acts on a metric measured space X with a negatively curved tangent bundle, then G acts on some Lp space, and that this action is proper under suitable assumptions. We then check that this result applies to the case when X is Gromov-hyperbolic.
We consider horofunction compactifications of symmetric spaces with respect to invariant Finsler metrics. We show that any (generalized) Satake compactification can be realized as a horofunction compactification with respect to a polyhedral Finsler metric.
In this article, we show that the Goldman-Iwahori metric on the space of all norms on a fixed vector space satisfies the Helly property for balls.
On the non-Archimedean side, we deduce that most classical Bruhat-Tits buildings may be endowed with a natural piecewise ℓ∞ metric which is injective. We also prove that most classical semisimple groups over non-Archimedean local fields act properly and cocompactly on Helly graphs. This gives another proof of biautomaticity for their uniform lattices.
On the Archimedean side, we deduce that most classical symmetric spaces of non-compact type may be endowed with a natural invariant Finsler metric, restricting to an ℓ∞ norm on each flat, which is coarsely injective. We also prove that most classical semisimple groups over Archimedean local fields act properly and cocompactly on injective metric spaces. We identify the injective hull of the symmetric space of GL(n,R) as the space of all norms on R^n. The only exception is the special linear group: if n=3 or n is at least 5 and K is a local field, we show that SL(n,K) does not act properly and coboundedly on an injective metric space.
We relate three classes of nonpositively curved metric spaces: hierarchically hyperbolic spaces, coarsely injective spaces, and strongly shortcut spaces. We show that any hierarchically hyperbolic space admits a new metric that is coarsely injective. The new metric is quasi-isometric to the original metric and is preserved under automorphisms of the hierarchically hyperbolic space. We show that any coarsely injective metric space of uniformly bounded geometry is strongly shortcut. Consequently, hierarchically hyperbolic groups---including mapping class groups of surfaces---are coarsely injective and coarsely injective groups are strongly shortcut. Using these results we deduce several important properties of hierarchically hyperbolic groups, including that they are semihyperbolic, have solvable conjugacy problem, are of type FP∞, have finitely many conjugacy classes of finite subgroups, and their finitely generated abelian subgroups are undistorted. Along the way we show that hierarchically quasiconvex subgroups of hierarchically hyperbolic groups have bounded packing.
We describe a simple locally CAT(0) classifying space for extra extra large type Artin groups (with all labels at least 5). Furthermore, when the Artin group is not dihedral, we describe a rank 1 periodic geodesic, thus proving that extra extra large type Artin groups are acylindrically hyperbolic. Together with Property RD proved by Ciabonu, Holt and Rees, the CAT(0) property implies the Baum-Connes conjecture for all extra extra large type Artin groups.
We prove that some classes of triangle-free Artin-Tits groups act properly on locally finite, finite-dimensional CAT(0) cube complexes. In particular, this provides the first examples of Artin-Tits 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.
We give a conjectural classification of virtually cocompactly cubulated Artin-Tits groups (i.e. having a finite index subgroup acting geometrically on a CAT(0) cube complex), which we prove for all Artin-Tits groups of spherical type, FC type or two-dimensional type. A particular case is that for n at least 4, the n-strand braid group is not virtually cocompactly cubulated.
We prove that any action of a higher rank lattice on a Gromov-hyperbolic space is elementary. More precisely, it is either elliptic or parabolic. This is a large generalization of the fact that any action of a higher rank lattice on a tree has a fixed point. A consequence is that any quasi-action of a higher rank lattice on a tree is elliptic, i.e. it has Manning's property (QFA). Moreover, we obtain a new proof of the theorem of Farb-Kaimanovich-Masur that any morphism from a higher rank lattice to a mapping class group has finite image, without relying on the Margulis normal subgroup theorem nor on bounded cohomology. More generally, we prove that any morphism from a higher rank lattice to a hierarchically hyperbolic group has finite image. In the Appendix, Vincent Guirardel and Camille Horbez deduce rigidity results for morphisms from a higher rank lattice to various outer automorphism groups.
We show that symmetric spaces and thick affine buildings which are not of spherical type A_1^r have no coarse median in the sense of Bowditch. As a consequence, they are not quasi-isometric to a CAT(0) cube complex, answering a question of Haglund. Another consequence is that any lattice in a simple higher rank group over a local field is not coarse median.
We show that braid groups with at most 6 strands are CAT(0) using the close connection between these groups, the associated non-crossing partition complexes and the embeddability of their diagonal links into spherical buildings of type A. Furthermore, we prove that the orthoscheme complex of any bounded graded modular complemented lattice is CAT(0), giving a partial answer to a conjecture of Brady and McCammond.
Let X be a symmetric space of non-compact type or a locally finite, strongly transitive Euclidean building, and let B denote the geodesic boundary of X. We reduce the study of visual limits of maximal flats in X to the study of limits of apartments in the spherical building B: this defines a natural, geometric compactification of the space of maximal flats of X. We then completely determine the possible degenerations of apartments when X is of rank 1, associated to a classical group of rank 2 or to PGL(4). In particular, we exhibit remarkable behaviours of visual limits of maximal flats in various symmetric spaces of small rank and surprising algebraic restrictions that occur.
We define a compactification of symmetric spaces of noncompact type, seen as spaces of isometry classes of marked lattices, analogous to the Thurston compactification of the Teichm\"uller space, and we show that it is equivariantly isomorphic to a Satake compactification. We then use it to define a new compactification of the Torelli space of a hyperbolic surface with marked points, and we show that it is equivariantly isomorphic to the Satake compactification of the image of the period mapping. Finally, we describe the natural stratification of a subset of the boundary.
Let G be a real semisimple Lie group with finite center, with a finite number of connected components and without compact factor. We are interested in the homogeneous space of Cartan subgroups of G, which can be also seen as the space of maximal flats of the symmetric space of G. We define its Chabauty compactification as the closure in the space of closed subgroups of G, endowed with the Chabauty topology. We show that when the real rank of G is 1, or when G=SL3(R) or SL4(R), this compactification is the set of all closed connected abelian subgroups of dimension the real rank of G, with real spectrum. And in the case of SL3(R), we study its topology more closely and we show that it is simply connected.
The space of closed subgroups of a locally compact topological group is endowed with a natural topology, called the Chabauty topology. Let X be a symmetric space of noncompact type, and G be its group of isometries. The space X identifies with the subspace of maximal compact subgroups of G: taking the closure gives rise to the Chabauty compactification of the symmetric space X. Using simpler arguments than those present in the book of Guivarc'h, Ji and Taylor, we describe the subgroups that appear in the boundary of the compactification.
The space of closed subgroups of a locally compact topological group is endowed with a natural topology, called the Chabauty topology. We completely describe the space of closed sugroups of the group RxZ, which is not trivial : for example, its fundamental group is uncountable.
Consider an affine Coxeter group W acting by isometries on the Euclidean space R^n, and the arrangement of its reflection hyperplanes. The fundamental group of the complement Y_W of the complexification of this arrangement in C^n mod out by W is the affine Artin group G_W associated with W. The K(π,1) conjecture states that Y_W is a classifying space for G_W. It has been recently proved by Paolini and Salvetti building on the works of McCammond and Sulway. We will present some ingredients of the proof that rests on the study of dual Garside structures for affine Artin groups, the factorisations of Euclidean isometries, and the shellability of noncrossing partitions. One consequence is that affine Artin groups, as well as braided crystallographic groups, have a finite classifying space.