arXiv subjects

Rana Sardar

Publications and source records attributed to Rana Sardar.

4 recordsLinked to original sources

arXiv2026

On 2-Sphere Bowditch Boundaries Attaining Conformal Dimension

Bonk and Kleiner proved that if $G$ is a Gromov hyperbolic group whose boundary $\partial_{\infty}G$ is homeomorphic to an Ahlfors $Q$-regular metric $2$-sphere $Z$, and the Ahlfors regular conformal dimension of $Z$ is attained and equal to $Q$, then $G$ acts discretely, cocompactly, and isometrically on $\mathbb{H}^3$. In this article, we extend the Bonk-Kleiner theorem to the setting of relatively hyperbolic groups. More precisely, we prove that if $(G,\mathcal{H})$ is a relatively hyperbolic group whose Bowditch boundary is homeomorphic to an Ahlfors $Q$-regular metric $2$-sphere $Z$, with the Ahlfors regular conformal dimension of $Z$ attained and equal to $Q$, then $G$ acts discretely and isometrically on $\mathbb{H}^3$, and every subgroup in $\mathcal{H}$ is virtually $\mathbb Z^2$.

math.GR↗

arXiv2026

Quasiconformal Maps between Bowditch Boundaries of Relatively Hyperbolic Groups

Classifying finitely generated groups up to quasi-isometry is a central problem in geometric group theory. In the context of hyperbolic and relatively hyperbolic groups, one of the key invariants in this classification is the boundary at infinity. Frédéric Paulin proved that two hyperbolic groups are quasi-isometric if and only if their Gromov boundaries are quasiconformally equivalent. In this article, we extend this correspondence to relatively hyperbolic groups via their Bowditch boundaries. We introduce a notion of quasiconformal maps on Bowditch boundaries that coarsely preserve shadows of horoballs relative to boundary points. We prove that any coarsely cusp-preserving quasi-isometry between relatively hyperbolic groups induces such a quasiconformal boundary map. Conversely, we prove that every quasiconformal homeomorphism of Bowditch boundaries that coarsely preserves shadows of horoballs arises from a coarsely cusp-preserving quasi-isometry between the groups.

math.GT↗

arXiv2026

A Characterization of Relative Hyperbolicity via Morse and Contracting Boundaries

We prove the following boundary-theoretic characterization of relatively hyperbolic groups. Let $G$ be a finitely generated group with a finite collection $\mathcal{H}$ of finitely generated subgroups, and let $G^h$ denote the associated cusped space. We prove that the pair $(G,\mathcal{H})$ is non-elementary relatively hyperbolic if and only if the Morse boundary $\partial_M^{\mathcal{DL}} G^h$ or the contracting boundary $\partial_c^{\mathcal{FQ}} G^h$ is non-empty and compact.

math.GT↗

arXiv2026

Maps between Boundaries of Relatively Hyperbolic Groups

F. Paulin proved that if the Gromov boundaries of two hyperbolic groups are quasi-Mobius equivalent, then the groups themselves are quasi-isometric. The goal of this article is to extend Paulin's result to the setting of relatively hyperbolic groups by introducing the notion of relative quasi-Mobius maps between the Bowditch boundaries of relatively hyperbolic groups. We show that any coarsely cusp-preserving quasi-isometry between two relatively hyperbolic groups induces a homeomorphism between their Bowditch boundaries, and that this induced homeomorphism is relative quasi-Mobius and linearly distorts the exit points of bi-infinite geodesics into combinatorial horoballs. Conversely, we prove that if a homeomorphism between the Bowditch boundaries of two relatively hyperbolic groups preserves parabolic fixed points and is either relative quasi-Mobius or linearly distorts the exit points of bi-infinite geodesics into combinatorial horoballs, then it arises from a coarsely cusp-preserving quasi-isometry between the groups.

math.GT↗