Zariski Dense Subgroups, Number Theory and Geometric Applications

This book presents selected chapters from the program Zariskiâ"Dense Subgroups, Number Theory, and Geometric Applications, held at the International Center for Theoretical Sciences (ICTS) in Bengaluru, Karnataka, India, from January 1â"12, 2024. The program encompassed a rich array of topics centered around Zariski-dense subgroups, with connections to algebraic and Lie groups, geometry, and number theory. It highlights the application of Diophantine approximation techniques to questions on linear groups with bounded generation, as well as innovative developments in the Bruhatâ"Tits theory for algebraic groups over local fields. These ideas were explored through four mini-courses alongside numerous research and expository lectures. Chapters are published in two volumes: Volume 1 features expanded notes from four mini-courses and two expository talks, while Volume 2 comprises twelve original research articles. Collectively, the volumes make recent advances in the theory of Zariski-dense subgroups accessible to a broad mathematical audience. The topic has continued to draw significant interest, building on discussions from earlier meetings such as the MSRI workshop in Berkeley (2012) and the IPAM workshop at UCLA (2015). Over the past two decades, Zariski-dense subgroups of algebraic groups have become a focal point of intense research, yielding a wealth of results with far-reaching applications. Notably, this line of inquiry has contributed to the construction of expander graphs and the study of spectral gaps, developments that culminated in the theory of superstrong approximation.