Stationary Random Subgroups and Injectivity Radius of Hyperbolic Manifolds

Explore a 40-minute conference talk delivered at the Workshop on "Geometric and Asymptotic Group Theory with Applications 2023 - Groups and Dynamics" held at the Erwin Schrödinger International Institute for Mathematics and Physics (ESI). Delve into the application of probabilistic methods in solving geometric problems, focusing on hyperbolic manifolds and rank one locally symmetric spaces. Discover how the bottom of the spectrum of the Laplacian on these spaces relates to points with arbitrarily large injectivity radius. Compare this result to recent findings in higher rank locally symmetric spaces. Examine the probabilistic underpinnings of the proof, particularly the properties of non-free stationary actions of G and their stabilizers. Gain insights into joint work with Arie Levit, exploring discrete stabilizers, full limit sets, and exponential growth rates in relation to random walk entropy and drift.