Overview #

My undergraduate research project focuses on studying iterated monodromy groups (IMGs) of expanding Thurston maps, under the supervision of Prof. Zhiqiang Li (PKU) and Prof. Tianyi Zheng) (UCSD). This project lies at the intersection of dynamical systems, geometric group theory, and probability. Here is a slide that was presented during my undergraduate research thesis defense, providing a detailed overview of this research project.

Key Results #

We proved that the random walk on the infinite Schreier graph of an IMG of an expanding Thurston map is almost surely recurrent. This result was achieved through:

• Constructing geometric realizations of finite-level Schreier graphs using tile structures
• Extending these realizations to infinite levels via solenoids
• Applying Benjamini-Schramm’s theory on local convergence of planar graphs

Methods and Techniques #

Our approach combines various mathematical tools:

• Tile structure analysis of expanding Thurston maps
• Solenoid theory from Lyubich and Minsky’s work on laminations
• Geometric measure theory for compatible measures on solenoids
• Local convergence theory of planar graphs

Current Status #

The project is ongoing, with our immediate goals including:

• Strengthening the recurrence result
• Investigating the amenability of these groups
• Exploring connections to low-dimensional topology