Can one hear the shape of a group?

Abstract

The iso-spectrum problem for marked lengnth spectrum for Riemannian manifolds of negative curvature has a rich history. We rephrased the problems for metrics on discrete groups, discussed its connection to a conjecture by Margulis, and proved some results for “total relatively hyperbolic groups” in Koji Fujiwara, Journal of Topology and Analysis, 7(2), 345-359 (2015). This is a note from my talk on that paper and mainly discuss the connection between Riemannian geometry and group theory, and also some questions.