Hyungryul Baik, Limits of Canonical Metrics in Low-dimensions

For a tower of finite normal covers of graphs or surfaces, one can consider a sequence of metrics on the base given by pull-back of canonical metric of the covers. We show that such a sequence has a limit and it depends only on the cover approximated by the tower up to scaling. The case of compact Riemann surface where the tower approximates the universal cover is due to Kazhdan. In this talk, I will explain why one might be interested in such a problem, give a brief introduction to the main tool – L²-theory, and discuss some future directions. This talk is based on a joint work with Farbod Shokrieh and Chenxi Wu.