Explore a comprehensive lecture on "Heights on Character Varieties, Free Subgroups And Spectral Gaps" delivered by Emmanuel Breuillard as part of the "Zariski Dense Subgroups, Number Theory and Geometric Applications" program. Delve into advanced topics in algebraic and analytic number theory, arithmetic theory of algebraic groups, and their applications to geometry and topology. Gain insights into recent developments in the study of Zariski-dense subgroups, including bounded generation, stability, and asymptotic cohomology. Examine the connections between character varieties, free subgroups, and spectral gaps, and their implications for various mathematical disciplines. This 1-hour 21-minute talk, organized by the International Centre for Theoretical Sciences, offers a unique opportunity to engage with cutting-edge research in the field and explore open problems for future investigation.