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 sup 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 piecewise ℓ∞ metric which is coarsely Helly. We also prove that most classical semisimple groups over Archimedean local fields act properly and cocompactly on injective metric spaces.
The only exception is the special linear group: if n is at least 3 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 Helly spaces, and strongly shortcut spaces. We show that any hierarchically hyperbolic space admits a new metric that is coarsely Helly. 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 Helly metric space of uniformly bounded geometry is strongly shortcut. Consequently, hierarchically hyperbolic groups---including mapping class groups of surfaces---are coarsely Helly and coarsely Helly 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 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.
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.