"Mathematicians David Fisher, Dmitry Kleinbock, and Gregory Soifer highlight in this edited collection the foundations and evolution of research by mathematician Gregory Margulis. Margulis is unusual in the degree to which his solutions to particular problems have opened new vistas of mathematics. Margulis' ideas were central, for example, to developments that led to the recent Fields Medals of Elon Lindenstrauss and Maryam Mirzhakhani. The broad goal of this volume is to introduce these areas, their development, their use in current research, and the connections between them. The foremost experts on the topic have written each of the chapters in this volume with a view to making them accessible by graduate students and by experts in other parts of mathematics"--.

• Contents
• Introduction | David Fisher
• Part I. Arithmeticity, superrigidity, normal subgroups
  • 1. Superrigidity, arithmeticity, normal subgroups: results, ramifications, and directions | David Fisher
  • 2. An extension of Margulis’s superrigidity theorem | Uri Bader and Alex Furman
  • 3. The normal subgroup theorem through measure rigidity | Aaron Brown, Federico Rodriguez Hertz, and Zhiren Wang
• Part II. Discrete subgroups
  • 4. Proper actions of discrete subgroups of affine transformations | Jeffrey Danciger, Todd A. Drumm, William M. Goldman, and Ilia Smilga
  • 5. Maximal subgroups of countable groups: a survey | Tsachik Gelander, Yair Glasner, and Gregory Soifer
• Part III. Expanders, representations, spectral theory
  • 6. Tempered homogeneous spaces II | Yves Benoist and Toshiyuki Kobayashi
  • 7. Expansion in simple groups | Emmanuel Breuillard and Alexander Lubotzky
  • 8. Elements of a metric spectral theory | Anders Karlsson
• Part IV. Homogeneous dynamics
  • 9. Quantitative nondivergence and Diophantine approximation on manifolds | Victor Beresnevich and Dmitry Kleinbock
  • 10. Margulis functions and their applications | Alex Eskin and Shahar Mozes
  • 11. Recent progress on rigidity properties of higher rank diagonalizable actions and applications | Elon Lindenstrauss
  • 12. Effective arguments in unipotent dynamics | Manfred Einsiedler and Amir Mohammadi
  • 13. Effective equidistribution of closed hyperbolic subspaces in congruence quotients of hyperbolic spaces | Manfred Einsiedler and Philipp Wirth
  • 14. Dynamics for discrete subgroups of SL2(C) | Hee Oh