Watch a 49-minute conference talk delivered by Bruno Martelli from the University of Pisa at the Instituto de Matemática Pura e Aplicada, focusing on hyperbolic geometry and the relationship between fibers and monodromy. Part of a larger conference on hyperbolic geometry and group theory organized in partnership with the Clay Mathematics Institute and NSF, this presentation contributes to discussions about recent developments in hyperbolic geometry, hyperbolic group theory, and related areas. The talk is one of twenty featured presentations from international mathematicians, preceding a workshop week featuring three minicourses aimed at graduate students for in-depth exploration of key topics in hyperbolic manifolds and geometry.