Abdul Zalloum
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View article: Stable cylinders and fine structures for hyperbolic groups and curve graphs
Stable cylinders and fine structures for hyperbolic groups and curve graphs Open
In 1995, Rips and Sela asked if torsionfree hyperbolic groups admit globally stable cylinders. We establish this property for all residually finite hyperbolic groups and curve graphs of finite-type surfaces. These cylinders are fine object…
View article: Morse subsets of injective spaces are strongly contracting
Morse subsets of injective spaces are strongly contracting Open
We show that a quasi-geodesic in an injective metric space is Morse if and only if it is strongly contracting. Since mapping class groups and, more generally, hierarchically hyperbolic groups act properly and coboundedly on injective metri…
View article: Growth tightness and genericity for word metrics from injective spaces
Growth tightness and genericity for word metrics from injective spaces Open
Mapping class groups are known to admit geometric (proper, cobounded) actions on injective spaces. Starting with such an action, and relying only on geometric arguments, we show that all finite generating sets resulting from taking large e…
View article: Constructing metric spaces from systems of walls
Constructing metric spaces from systems of walls Open
We give a general procedure for constructing metric spaces from systems of partitions. This generalises and provides analogues of Sageev's construction of dual CAT(0) cube complexes for the settings of hyperbolic and injective metric space…
View article: Effective flipping, skewering and rank rigidity for cubulated groups with factor systems
Effective flipping, skewering and rank rigidity for cubulated groups with factor systems Open
Relying on work of Caprace and Sageev \cite{capracesageev:rank}, we provide an effective form of rank rigidity in the context of groups virtually acting freely cocompactly on a CAT(0) cube complex with a factor system. We accomplish this b…
View article: Uniform undistortion from barycentres, and applications to hierarchically hyperbolic groups
Uniform undistortion from barycentres, and applications to hierarchically hyperbolic groups Open
We show that infinite cyclic subgroups of groups acting uniformly properly on injective metric spaces are uniformly undistorted. In the special case of hierarchically hyperbolic groups, we use this to study translation lengths for actions …
View article: Injectivity, cubical approximations and equivariant wall structures beyond CAT(0) cube complexes
Injectivity, cubical approximations and equivariant wall structures beyond CAT(0) cube complexes Open
This is an expository survey with two goals. 1) The primary goal is to discuss and highlight the impact of two recent influential ideas in geometric group theory. The first of which is the notion of an injective metric space which is a ric…
View article: Sublinearly Morse boundaries from the viewpoint of combinatorics
Sublinearly Morse boundaries from the viewpoint of combinatorics Open
We prove that the sublinearly Morse boundary of CAT ( 0 ) {\mathrm{CAT}(0)} cubulated groups with factor systems continuously injects in the Gromov boundary of a certain hyperbolic graph Γ. We also show that for all CAT ( 0 ) …
View article: Morse subsets of injective spaces are strongly contracting
Morse subsets of injective spaces are strongly contracting Open
We show that a quasi-geodesic in an injective metric space is Morse if and only if it is strongly contracting. Since mapping class groups and, more generally, hierarchically hyperbolic groups act properly and coboundedly on injective metri…
View article: Sublinearly Morse geodesics in CAT(0) spaces: lower divergence and hyperplane characterization
Sublinearly Morse geodesics in CAT(0) spaces: lower divergence and hyperplane characterization Open
We introduce the notion of k-lower divergence for geodesic rays in CAT(0) spaces. Building on the work of Charney and Sultan we give various characterizations of k-contracting geodesic rays using k-lower divergence and k-slim triangles. We…
View article: Hyperbolic models for CAT(0) spaces
Hyperbolic models for CAT(0) spaces Open
We introduce two new tools for studying CAT(0) spaces: \emph{curtains}, versions of cubical hyperplanes; and the \emph{curtain model}, a counterpart of the curve graph. These tools shed new light on CAT(0) spaces, allowing us to prove a di…
View article: The geometry of genericity in mapping class groups and Teichmüller spaces via CAT(0) cube complexes
The geometry of genericity in mapping class groups and Teichmüller spaces via CAT(0) cube complexes Open
Random walks on spaces with hyperbolic properties tend to sublinearly track geodesic rays which point in certain hyperbolic-like directions. Qing-Rafi-Tiozzo recently introduced the sublinearly Morse boundary and proved that this boundary …
View article: Sublinearly Morse boundaries from the viewpoint of combinatorics
Sublinearly Morse boundaries from the viewpoint of combinatorics Open
We prove that the sublinearly Morse boundary of every known cubulated group continuously injects in the Gromov boundary of a certain hyperbolic graph. We also show that for all CAT(0) cube complexes, convergence to sublinearly Morse geodes…
View article: Convergence of sublinearly contracting horospheres
Convergence of sublinearly contracting horospheres Open
In \cite{QR19}, Qing, Rafi and Tiozzo 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 disjoin…
View article: Regularity of Morse geodesics and growth of stable subgroups
Regularity of Morse geodesics and growth of stable subgroups Open
We prove that Morse local-to-global groups grow exponentially faster than their infinite index stable subgroups. This generalizes a result of Dahmani, Futer, and Wise in the context of quasi-convex subgroups of hyperbolic groups to a broad…
View article: Rank One Isometries in Sublinearly Morse Boundaries of CAT(0) Groups.
Rank One Isometries in Sublinearly Morse Boundaries of CAT(0) Groups. Open
Given a sublinear function $\kappa$, the $\kappa$-Morse boundary $\partial_\kappa G$ of a CAT(0) group was introduced by Qing and Rafi and shown to be a quasi-isometry invariant and a metrizable space.
In this paper, we prove several pro…
View article: Geometry and dynamics on sublinearly Morse boundaries of CAT(0) groups
Geometry and dynamics on sublinearly Morse boundaries of CAT(0) groups Open
Given a sublinear function $κ$, $κ$-Morse boundaries $\pka X$ of proper \CAT spaces are introduced by Qing, Rafi and Tiozzo. It is a topological space that consists of a large set of quasi-geodesic rays and it is quasi-isometrically invari…
View article: Regular Languages for Contracting Geodesics
Regular Languages for Contracting Geodesics Open
Let $G$ be a finitely generated group. We show that for any finite generating set $A$, the language consisting of all geodesics in $Cay(G,A)$ with a contracting property is a regular language. As an application, we show that any finitely g…