Searcharxiv⌕ Search

arXiv subjects

Abdul Zalloum

Publications and source records attributed to Abdul Zalloum.

16 recordsLinked to original sources
arXiv2026

Constructing metric spaces from systems of walls

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 spaces. As applications, we produce a ``universal'' hyperbolic action for groups with strongly contracting elements, and show that many groups with ``coarsely cubical'' features admit geometric actions on injective metric spaces. In an appendix with Davide Spriano, we show that a large class of groups have an infinite-dimensional space of quasimorphisms.

math.GR↗
arXiv2026

Pseudo-Anosov flows and the geometry of Anosov-like group actions

We show that the action on its orbit space induced by a pseudo-Anosov flow on a closed $3$-manifold (and more general Anosov-like actions) can be seen as an isometric action on a Gromov-hyperbolic space. When the flow is not $\R$-covered, we show that this action admits elements that are weakly properly discontinuous and deduce that elements of $π_1(M)$ that do \emph{not} represent a periodic orbit of the flow are generic for any word metric coming from a finite generating set. We also give a number of other geometric group-theoretic results for Anosov-like group actions on bifoliated planes.

math.DS↗
arXiv2025

Growth tightness and genericity for word metrics from injective spaces

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 enough balls in the respective injective space yield word metrics where pseudo-Anosov maps are exponentially generic. We also show that growth tightness holds true for the Cayley graphs corresponding to these finite generating sets, providing a positive answer to a question by Arzhantseva, Cashen and Tao.

math.GT↗
arXiv2025

Uniform undistortion from barycentres, and applications to hierarchically hyperbolic groups

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 on the associated hyperbolic spaces. We then use quasimorphisms to produce examples where these latter results are sharp.

math.GT↗
arXiv2025

Stable cylinders and fine structures for hyperbolic groups and curve graphs

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 objects, and the core of our approach is to upgrade the hyperbolic space to one with improved fine properties via a generalisation of Sageev's construction. The methods also let us prove that curve graphs of surfaces admit equivariant quasiisometric embeddings in finite products of quasitrees.

math.GT↗
arXiv2024

Hyperbolic models for CAT(0) spaces

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 dichotomy of a rank-rigidity flavour, establish Ivanov-style rigidity theorems for isometries of the curtain model, find isometry-invariant copies of its Gromov boundary in the visual boundary of the underlying CAT(0) space, and characterise rank-one isometries both in terms of their action on the curtain model and in terms of curtains. Finally, we show that the curtain model is universal for WPD actions over all groups acting properly on the CAT(0) space.

math.MG↗
arXiv2023

Effective flipping, skewering and rank rigidity for cubulated groups with factor systems

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 by exhibiting a special pair of hyperplanes that can be skewered uniformly quickly. Furthermore, for virtually special compact groups, we prove an effective omnibus theorem and provide a trichotomy implying a strong form of an effective Tits alternative. More generally, we provide a recipe for producing short Morse elements generating free stable subgroups in any virtually torsion-free hierarchically hyperbolic group (HHG) which recovers Mangahas' work \cite{MangahasRecipie} and provides an effective rank-rigidity dichotomy in the context of HHGs via Durham-Hagen-Sisto \cite{Durham2017-ce}. Part of our analysis involves showing that Caprace-Sageev's cubical tools of flipping and skewering can be applied to any HHG using the notion of a \emph{curtain} recently introduced by Petyt, Spriano and the author in \cite{PSZCAT}, \cite{Zalloum23_injectivity}. Indeed, our route to producing a short Morse element proceeds by exhibiting a special pair of curtains in the underlying HHG that can be skewered uniformly quickly.

math.GT↗
arXiv2023

Injectivity, cubical approximations and equivariant wall structures beyond CAT(0) cube complexes

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 rich class of spaces that was imported to geometric group theory by Lang and have shown to be of a great effect. The second is Behrstock-Hagen-Sisto's cubical approximation theorem which provides a novel and particularly successful approach for studying mapping class groups (of finite type surfaces) and more generally, hierarchically hyperbolic groups. 2) Our second goal is to demonstrate how numerous geodesic metric spaces including hyperbolic spaces, CAT(0) spaces, and hierarchically hyperbolic spaces admit a strikingly rich equivariant wall structure: a discovery that was inspired by the aforementioned machines; the cubical approximation theorem as well as injective metric spaces.

math.GR↗
arXiv2023

Morse subsets of injective spaces are strongly contracting

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 metric spaces, we deduce various consequences relating, for example, to growth tightness and genericity of pseudo-Anosovs/Morse elements. Moreover, we show that injective metric spaces have the Morse local-to-global property and that a non-virtually-cyclic group acting properly and coboundedly on an injective metric space is acylindrically hyperbolic if and only it contains a Morse ray.

math.GT↗
arXiv2023

Geometry and dynamics on sublinearly Morse boundaries of CAT(0) groups

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 invariant and metrizable. In this paper, we study the sublinearly Morse boundaries with the assumption that there is a proper cocompact action of a group $G$ on the \CAT space in question. We show that $G$ acts minimally on $\pka G$ and that contracting elements of $G$ induces a weak north-south dynamic on $\pka G$. Furthermore, we show that a homeomorphism $f \from \pka G \to \pka G'$ comes from a quasi-isometry if and only if $f$ is successively quasi-m{ö}bius and stable. Lastly, we characterize exactly when the sublinearly Morse boundary of a \CAT space is compact.

math.GR↗
arXiv2022

The geometry of genericity in mapping class groups and Teichmüller spaces via CAT(0) cube complexes

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 is a quasi-isometry invariant which captures a notion of generic direction in a broad context. In this article, we develop the geometric foundations of sublinear Morseness in the mapping class group and Teichmüller space. We prove that their sublinearly Morse boundaries are visibility spaces and admit continuous equivariant injections into the boundary of the curve graph. Moreover, we completely characterize sublinear Morseness in terms of the hierarchical structures of these spaces. Our techniques include developing tools for modeling the hulls of median rays in hierarchically hyperbolic spaces via CAT(0) cube complexes. Part of this analysis involves establishing direct connections between the geometry of the curve graph and the combinatorics of hyperplanes in the approximating cube complexes.

math.GT↗
arXiv2022

Regular Languages for Contracting Geodesics

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 generated group containing an infinite contracting geodesic must be either virtually $\mathbb{Z}$ or acylindrically hyperbolic.

math.GR↗
arXiv2022

Regularity of Morse geodesics and growth of stable subgroups

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 class of groups that contains the mapping class group, CAT(0) groups, and the fundamental groups of closed 3-manifolds. To accomplish this, we develop a theory of automatic structures on Morse geodesics in Morse local-to-global groups. Other applications of these automatic structures include a description of stable subgroups in terms of regular languages, rationality of the growth of stable subgroups, density in the Morse boundary of the attracting fixed points of Morse elements, and containment of the Morse boundary inside the limit set of any infinite normal subgroup.

math.GR↗
arXiv2022

Convergence of sublinearly contracting horospheres

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 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 complete 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.

math.GT↗
arXiv2021

Sublinearly Morse boundaries from the viewpoint of combinatorics

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 geodesic rays has a simple combinatorial description using the hyperplanes crossed by such sequences. As an application of this combinatorial description, we show that a certain subspace of the Roller boundary continously surjects on the subspace of the visual boundary consisting of sublinearly Morse geodesic rays.

math.GT↗