WoG 2026 Talk 3.6: Rana Sardar - On 2-sphere Bowditch boundaries attaining conformal dimension

Speaker: Rana Sardar Institution: IISER Mohali Title: On 2-sphere Bowditch boundaries attaining conformal dimension Abstract: The relative Cannon conjecture predicts that a relatively hyperbolic group whose Bowditch boundary is homeomorphic to the standard 2-sphere is Kleinian, with virtually $\mathbb{Z}^2$ peripheral subgroups. In this talk, I will establish this conjecture under an additional analytic assumption on the boundary. I will begin by recalling the basic notions of a relatively hyperbolic group $(G,\mathcal{P})$ and its Bowditch boundary, followed by the relevant metric notions: Ahlfors $Q$-regularity and the Ahlfors regular conformal dimension. I will then discuss the background to the problem, including the classical Cannon conjecture for hyperbolic groups and its analytic formulation due to Bonk and Kleiner. Finally, I will present the main result: if $(G,\mathcal{P})$ is relatively hyperbolic, its Bowditch boundary is homeomorphic to the standard 2-sphere, and the boundary admits an Ahlfors $Q$-regular metric attaining the Ahlfors regular conformal dimension, then $G$ acts discretely and isometrically on hyperbolic 3-space $\mathbb{H}^3$, and every peripheral subgroup $P \in \mathcal{P}$ is virtually $\mathbb{Z}^2$.