arXiv subjects

Hussain AL-Rasheed

Publications and source records attributed to Hussain AL-Rasheed.

2 recordsLinked to original sources

arXiv2026

Ends of Uncountable Locally Finite Groups: An Analytic and Coarse Geometric Approach

It is a classical result of D. F. Holt that an uncountable, locally finite discrete group possesses exactly one end. While traditional approaches rely on cohomological methods and the combinatorial manipulation of almost-invariant subsets, we establish this result through coarse geometry via the topological structure of the Higson corona. By interpreting the boundaries of coarsely clopen sets in terms of subgroup invariance, we evaluate their associated characteristic functions on the large-scale geometry of the group. Utilizing a Skolem-hull extraction and a periodic coarse rerouting technique, we force a geometric rigidity that prohibits multiple ends.

math.GR↗

arXiv2026

Coarsely Proper Actions of Topological Groups

We adapt the classical topological notions of properly discontinuous group actions to the framework of coarse geometry. By substituting the rigid constraints of local compactness and discreteness with uniform coarse equivalences, we resolve the topological obstructions identified by Kapovich. We establish large-scale characterisations for coarsely proper actions and coboundedness. As a primary application, we provide an affirmative answer to an open problem posed by Rosendal (Problem B.7) regarding the intrinsic coarse equivalence of cocompact subgroups in Polish groups. Furthermore, we construct large-scale fundamental sets via coarse Dirichlet domains, generalise the \v{S}varc-Milnor Lemma, and utilise the isometry group of the Urysohn universal space as an application.

math.GR↗