Arithmetic Groups - Advanced Topics in Rigidity, Cohomology, and Dynamics

Explore advanced topics in arithmetic groups through this comprehensive collection of research lectures from leading mathematicians at the Institute for Advanced Study. Delve into first-order rigidity of high-rank arithmetic groups, bi-interpretability and congruence subgroups, and bounded generation properties with concrete examples. Examine the congruence subgroup property for SL(2,Z), algebraicity and holonomicity theorems, and the intricate connections between commutators in SL_2 and Markoff surfaces. Investigate Grothendieck pairs and profinite rigidity, representation rigidity in profinite completions, and the transition from PSL2 representation rigidity to profinite rigidity. Study Anosov groups through the lens of local mixing, counting, and equidistribution, while exploring effective equidistribution of one-parameter unipotent flows with polynomial bounds. Learn about vanishing theorems for bounded cohomology as preparation for stability theory, asymptotic bounded cohomology, and uniform stability of high-rank lattices. Discover canonical forms for free group automorphisms, growth patterns of Bianchi modular forms, and arithmetic and dynamics on varieties of Markoff type. Conclude with an exploration of bounded generation through Diophantine geometry, providing a comprehensive overview of current research directions in arithmetic group theory and its applications to number theory, algebraic geometry, and dynamical systems.