On dynamical Kapovich-Kleiner conjecture

YMSC Topology Seminar

Organizers：

陈伟彦、高鸿灏、黄意、林剑锋、孙巍峰

Speaker：

Wen-Bo Li李文博

BICMR, Peking University

Time：

Thur., 16:00-17:00, Oct. 24, 2024

Venue：

B725, Shuangqing Complex Building

清华大学双清综合楼A座B725

Online：

Zoom Meeting ID: 405 416 0815

Passcode: 111111

Title：

On dynamical Kapovich-Kleiner conjecture

Abstract:

The Kapovich-Kleiner conjecture predicts that Gromov hyperbolic groups with Sierpiński carpet boundaries admit geometric actions on a convex subset of hyperbolic 3-space with a nonempty totally geodesic boundary. An equivalent version of this conjecture can be phrased as the Sierpiński carpet boundaries can be embedded into the Riemann sphere by a quasisymmetry.

Following Sullivan's dictionary, which draws analogies between group theory and dynamics, we present a dynamical analog of this conjecture using Thurston maps —branched coverings of the 2-sphere with finite postcritical sets. We characterize when subsystems of these maps, with tile maximal invariant sets homeomorphic to the Sierpiński carpet, are topologically conjugate to subsystems of postcritically-finite rational maps. This holds if and only if the invariant set can be quasisymmetrically embedded into the Riemann sphere, or equivalently, if there is no Thurston obstruction for the subsystem. Our results bridge uniformization in dynamics with conjectures in geometric group theory. This is a joint work with Mario Bonk (UCLA) and Zhiqiang Li (Peking University).