Conformal Dimension and the Quasisymmetric Geometry of Metric Spaces.

[摘要] The conformal dimension of a metric space measures the optimal dimension of the space under quasisymmetric deformations. We consider metric spaces that are locally connected and have no local cut points, in a quantitative way, and show that such spaces have conformal dimension greater than one.We then apply this result to hyperbolic groups that do not virtually split over any finite or virtually cyclic subgroup to show that the conformal dimension of the conformal boundary at infinity of such groups is greater than one. This answers a question of Bonk and Kleiner.One additional result is worth noting: we show that a linearly connected, doubling metric space is connected by quasi-arcs, quantitatively. While this was previously proven by Tukia, our proof is new and much improved.

[发布机构] University of Michigan

[效力级别] Quasisymmetric Maps

[关键词] Conformal Dimension;Quasisymmetric Maps;Hyperbolic Group;Boundary at Infinity;Hausdorff Dimension;No Local Cut Points;Mathematics;Science;Mathematics