Proper CAT Research Articles

Characterising trees and hyperbolic spaces by their boundaries

We use the language of proper CAT(-1) spaces to study thick, locally compact trees, the real, complex and quaternionic hyperbolic spaces and the hyperbolic plane over the octonions. These are rank 1 Euclidean buildings, respectively rank 1 symmetric spaces of non-compact type. We give a uniform proof that these spaces may be reconstructed using the cross ratio on their visual boundary, bringing together the work of Tits and Bourdon.

Seroprevalence and risk factors for Toxoplasma gondii infection among pregnant women at Debre Markos Referral Hospital, northwest Ethiopia.

To assess the seroprevalence and risk factors of Toxoplasma gondii infection in pregnant women at the Debre Markos Referral Hospital, northwest Ethiopia. A facility-based cross-sectional study was undertaken among pregnant women from March 2020 to May 2021. Sociodemographic and clinical data were collected from randomly selected participants. Five millilitres of blood was collected and an enzyme-linked immunosorbent assay kit was used to test for T. gondii immunoglobulin G (IgG) antibodies. A logistic regression model was computed to identify the risk factors. The adjusted odds ratio (aOR) was estimated along with the 95% confidence interval (CI). A statistically significant association was defined as p<0.05. T. gondii IgG antibody positivity was found in 38.8% (n=132) of 340 pregnant women. Contact with cats (AOR 2.5 [95% CI 1.5 to 4.2]), eating raw/undercooked meat (AOR 5.7 [95% CI 3.2 to 10.3]), consuming unwashed vegetables (AOR 4.1 [95% CI 2.1 to 8.0]), a history of abortion (AOR 1.9 [95% CI 1.1 to 3.3]) and drinking water sources (AOR 2.5 [95% CI 1.2 to 5.2]) demonstrated a statistically significant association with T. gondii infection. Toxoplasmosis was found to be fairly common in pregnant mothers. Proper cat excreta disposal, not eating raw/undercooked meat, maintaining hand cleanliness and following environmental sanitation protocols could be important to decrease T. gondii infection.

Convergence of sublinearly contracting horospheres

Qing and Rafi (Morse boundaries I: CAT(0) spaces, 2019. arXiv:1909.02096) introduced the sublinearly contracting boundary for CAT(0) spaces. Every point of this boundary is uniquely represented by a sublinearly contracting geodesic ray: a geodesic ray b where every disjoint ball projects to a subset whose diameter is bounded by a sublinear function in terms of the ball’s distance to the origin. This paper analyzes the bahaviour of horofunctions associated to such geodesic rays, for example, we show that horospheres associated to such horofunctions are convergent. As a consequence of this analysis, we show that for any proper CAT(0) space X, every point of the visual boundary \(\partial X\) that is defined by a sublinearly contracting geodesic ray is a visibility point.

Continua in the Gromov–Hausdorff space

We first prove that for all compact metrizable spaces, there exists a topological embedding of the compact metrizable space into each of the sets of compact metric spaces which are connected, path-connected, geodesic, or CAT(0), in the Gromov–Hausdorff space with finite prescribed values. As its application, we show that the sets prescribed above are path-connected and their non-empty open subsets have infinite topological dimension. By the same method, we also prove that the set of all proper CAT(0) spaces is path-connected and its non-empty open subsets have infinite topological dimension with respect to the pointed Gromov–Hausdorff distance.

Higher horospherical limit sets for G-modules over CAT(0)-spaces

AbstractThe Σ-invariants of Bieri–Neumann–Strebel and Bieri–Renz involve an action of a discrete group G on a geometrically suitable space M. In the early versions, M was always a finite-dimensional Euclidean space on which G acted by translations. A substantial literature exists on this, connecting the invariants to group theory and to tropical geometry (which, actually, Σ-theory anticipated). More recently, we have generalized these invariants to the case where M is a proper CAT(0) space on which G acts by isometries. The “zeroth stage” of this was developed in our paper [BG16]. The present paper provides a higher-dimensional extension of the theory to the “nth stage” for any n.

Boundary conditions detecting product splittings of CAT(0) spaces

Let X be a proper CAT(0) space and G a group of isometries of X acting cocompactly without fixed point at infinity. We prove that if \partial X contains an invariant subset of circumradius \pi/2 , then X contains a quasi-dense, closed convex subspace that splits as a product. Adding the assumption that the G -action on X is properly discontinuous, we give more conditions that are equivalent to a product splitting. In particular, this occurs if \partial X contains a proper nonempty, closed, invariant, \pi -convex set in \partial X ; or if some nonempty closed, invariant set in \partial X intersects every round sphere K \subset \partial X inside a proper subsphere of K .

On hierarchical hyperbolicity of cubical groups

Let X be a proper CAT(0) cube complex admitting a proper cocompact action by a group G. We give three conditions on the action, any one of which ensures that X has a factor system in the sense of [BHS14]. We also prove that one of these conditions is necessary. This combines with results of Behrstock--Hagen--Sisto to show that $G$ is a hierarchically hyperbolic group; this partially answers questions raised by those authors. Under any of these conditions, our results also affirm a conjecture of BehrstockHagen on boundaries of cube complexes, which implies that X cannot contain a convex staircase. The conditions on the action are all strictly weaker than virtual cospecialness, and we are not aware of a cocompactly cubulated group that does not satisfy at least one of the conditions.

On semistability of CAT(0) groups

Does every one-ended CAT(0) group have semistable fundamental group at infinity? As we write, this is an open question. Let G be such a group acting geometrically on the proper CAT(0) space X . In this paper we show that in order to establish a positive answer to the question it is only necessary to check that any two geodesic rays in X are properly homotopic. We then show that if the answer to the question is negative, with (G, X) a counter-example, then the boundary of X, \partial X with the cone topology, must have a weak cut point. This is of interest because a theorem of Papasoglu and the second-named author [10] has established that there cannot be an example of (G, X) where \partial X has a cut point. Thus, the search for a negative answer comes down to the difference between cut points and weak cut points. We also show that the Tits ball of radius \frac{\pi}{2} about that weak cut point is a “cut set” in the sense that it separates \partial X . Finally, we observe that if a negative example (G, X) exists then G is rank 1.

Topology and dynamics of the contracting boundary of cocompact CAT(0) spaces

Let $X$ be a proper CAT(0) space and let $G$ be a cocompact group of isometries of $X$ which acts properly discontinuously. Charney and Sultan constructed a quasi-isometry invariant boundary for proper CAT(0) spaces which they called the contracting boundary. The contracting boundary imitates the Gromov boundary for $\delta$-hyperbolic spaces. We will make this comparison more precise by establishing some well known results for the Gromov boundary in the case of the contracting boundary. We show that the dynamics on the contracting boundary is very similar to that of a $\delta$-hyperbolic group. In particular the action of $G$ on $\partial_cX$ is minimal if $G$ is not virtually cyclic. We also establish a uniform convergence result that is similar to the $\pi$-convergence of Papasoglu and Swenson and as a consequence we obtain a new north-south dynamics result on the contracting boundary. We additionally investigate the topological properties of the contracting boundary and we find necessary and sufficient conditions for $G$ to be $\delta$-hyperbolic. We prove that if the contracting boundary is compact, locally compact or metrizable, then $G$ is $\delta$-hyperbolic.

Local rigidity of uniform lattices

We establish topological local rigidity for uniform lattices in compactly generated groups, extending the result of Weil from the realm of Lie groups. We generalize the classical local rigidity theorem of Selberg, Calabi and Weil to irreducible uniform lattices in Isom (X) where X is a proper CAT(0) space with no Euclidian factors, not isometric to the hyperbolic plane. We deduce an analog of Wang’s finiteness theorem for certain non-positively curved metric spaces.

Proper homotopy types and [formula omitted]-boundaries of spaces admitting geometric group actions

We extend several techniques and theorems from geometric group theory so they apply to geometric actions on arbitrary proper metric ARs (absolute retracts). Previous versions often required actions on CW complexes, manifolds, or proper CAT(0) spaces, or else included a finite-dimensionality hypothesis. We remove those requirements, providing proofs that simultaneously cover all of the usual variety of spaces. A second way that we generalize earlier results is by eliminating a freeness requirement often placed on the group actions. In doing so, we allow for groups with torsion.The main theorems are new in that they generalize results found in the literature, but a significant aim is expository. Toward that end, brief but reasonably comprehensive introductions to the theories of ANRs (absolute neighborhood retracts) and Z-sets are included, as well as a much shorter introduction to shape theory. Here is a sampling of the theorems proved here.Theorem.If quasi-isometric groupsG andH act geometrically on proper metric ARsX andY, resp., thenX is proper homotopy equivalent toY.Theorem.If quasi-isometric groupsG andH act geometrically on proper metric ARsX andY, resp., andY can be compactified to aZ-structureY¯,Z forH, then the same boundary can be added toX to obtain aZ-structure forG.Theorem.If quasi-isometric groupsGandH admitZ-structuresX¯,Z1 andY¯,Z2, resp., thenZ1 andZ2 are shape equivalent.

Morse boundaries of proper geodesic metric spaces

We introduce a new type of boundary for proper geodesic spaces, called the Morse boundary, that is constructed with rays that identify the “hyperbolic directions” in that space. This boundary is a quasi-isometry invariant and thus produces a well-defined boundary for any finitely generated group. In the case of a proper CAT(0) space this boundary is the contracting boundary of Charney and Sultan, and in the case of a proper Gromov hyperbolic space this boundary is the Gromov boundary. We prove three results about the Morse boundary of Teichmüller space. First, we show that the Morse boundary of the mapping class group of a surface is homeomorphic to the Morse boundary of the Teichmüller space of that surface. Second, using a result of Leininger and Schleimer, we show that Morse boundaries of Teichmüller space can contain spheres of arbitrarily high dimension. Finally, we show that there is an injective continuous map of the Morse boundary of Teichmüller space into the Thurston compactication of Teichmüller space by projective measured foliations.

Limit sets for modules over groups onCAT(0) spaces: from the Euclidean to the hyperbolic

The observation that the 0-dimensional Geometric Invariant $\Sigma ^{0}(G;A)$ of Bieri-Neumann-Strebel-Renz can be interpreted as a horospherical limit set opens a direct trail from Poincar\'e's limit set $\Lambda (\Gamma)$ of a discrete group $\Gamma $ of M\"obius transformations (which contains the horospherical limit set of $\Gamma $) to the roots of tropical geometry (closely related to $\Sigma ^{0}(G;A)$ when G is abelian). We explore this trail by introducing the horospherical limit set, $\Sigma (M;A)$, of a G-module A when G acts by isometries on a proper CAT(0) metric space M. This is a subset of the boundary at infinity of M. On the way we meet instances where $\Sigma (M;A)$ is the set of all conical limit points, the complement of a spherical building, the complement of the radial projection of a tropical variety, or (via the Bieri-Neumann-Strebel invariant) where it is closely related to the Thurston norm.

On equivariant homeomorphisms of boundaries of CAT(0) groups and Coxeter groups

In this paper, we investigate an equivariant homeomorphism of the boundaries ∂X and ∂Y of two proper CAT(0) spaces X and Y on which a CAT(0) group G acts geometrically. We provide a sufficient condition and an equivalent condition to obtain a G-equivariant homeomorphism of the boundaries ∂X and ∂Y as a continuous extension of the quasi-isometry ϕ:Gx0→Gy0 defined by ϕ(gx0)=gy0, where x0∈X and y0∈Y. In this paper, we say that a CAT(0) group G is equivariant (boundary) rigid, if G determines its ideal boundary by the equivariant homeomorphisms as above. As an application, we introduce some examples of (non-)equivariant rigid CAT(0) groups and we show that if Coxeter groups W1 and W2 are equivariant rigid as reflection groups, then so is W1⁎W2. We also provide a conjecture on non-rigidity of boundaries of some CAT(0) groups.

Erratum to "Asymptotic dimension and boundary dimension of proper CAT(0) spaces"

The review on [1] in Mathematical Reviews points out that the proof of its main result is incorrect. The aim of this paper is to correct the previous paper's argument and clarify the statement.

An indiscrete Bieberbach theorem: from amenable CAT$(0)$ groups to Tits buildings

Non-positively curved spaces admitting a cocompact isometric action of an amenable group are investigated. A classification is established under the assumption that there is no global fixed point at infinity under the full isometry group. The visual boundary is then a spherical building. When the ambient space is geodesically complete, it must be a product of flats, symmetric spaces, biregular trees and Bruhat–Tits buildings. We provide moreover a sufficient condition for a spherical building arising as the visual boundary of a proper CAT(0) space to be Moufang, and deduce that an irreducible locally finite Euclidean building of dimension ≥ 2 is a Bruhat–Tits building if and only if its automorphism group acts cocompactly and chamber-transitively at infinity.

Logarithm laws for strong unstable foliations in negative curvature and non-Archimedian Diophantine approximation

Given a finite volume negatively curved Riemannian manifold M , we give a precise relation between the logarithmic growth rates of the excursions of the strong unstable leaves of negatively recurrent unit tangent vectors into cusp neighborhoods of M and their linear divergence rates under the geodesic flow. Our results hold in the more general setting where M is the quotient of any proper CAT(±1) metric space X by any geometrically finite discrete group of isometries of X . As an application to non-Archimedian Diophantine approximation in positive characteristic, we relate the growth of the orbits of \mathcal{O}_{\hat K} -lattices under one-parameter unipotent subgroups of \mathrm{GL}_2(\hat K) with approximation exponents and continued fraction expansions of elements of the local field \hat K of formal Laurent series over a finite field.

Controlled connectivity for semidirect products acting on locally finite trees

In 2003 Bieri and Geoghegan generalized the Bieri- Neuman-Strebel invariant � 1 by defining � 1 (�), � an isometric action by a finitely generated group G on a proper CAT(0) space M. In this paper, we show how the natural and well-known connec- tion between Bass-Serre theory and covering space theory provides a framework for the calculation of � 1 (�) whenis a cocompact action by G = B ⋊ A, A a finitely generated group, on a locally finite Bass-Serre tree T for A. This framework leads to a theorem providing conditions for including an endpoint in, or excluding an endpoint from, � 1 (�). When A is a finitely generated free group acting on its Cayley graph, we can restate this theorem from a more algebraic perspective, which leads to some general results on � 1 for such actions.

CAT(0) spaces with boundary the join of two Cantor sets

We will show that if a proper complete CAT(0) space X has a visual boundary homeomorphic to the join of two Cantor sets, and X admits a geometric group action by a group containing a subgroup isomorphic to Z^2, then its Tits boundary is the spherical join of two uncountable discrete sets. If X is geodesically complete, then X is a product, and the group has a finite index subgroup isomorphic to a lattice in the product of two isometry groups of bounded valence bushy trees.

On bounded cocycles of isometries over minimal dynamics

We show the following geometric generalization of a classical theoremof W. H. Gottschalk and G. A. Hedlund: a skew action induced by acocycle of (affine) isometries of a Hilbert space over a minimaldynamical system has a continuous invariant section if and only if the cocycleis bounded. Equivalently, the associated twisted cohomologicalequation has a continuous solution if and only if the cocycle isbounded. We interpret this as a version of the Bruhat-Tits CenterLemma in the space of continuous functions. Our result also holds whenthe fiber is a proper CAT(0) space. One of the applications concernsmatrix cocycles. Using the action of $\mathrm{GL} (n,\mathbb{R})$ onthe (nonpositively curved) space of positively definite matrices, weshow that every bounded linear cocycle over a minimal dynamical system iscohomologous to a cocycle taking values in the orthogonal group.