Commensurators of Thin Subgroups

Explore a 54-minute lecture on commensurators of thin subgroups delivered by Mahan M.J. at the International Centre for Theoretical Sciences. Delve into the intricate world of ergodic theory and dynamical systems, tracing their origins from Boltzmann's kinetic theory of gases to modern applications in mathematics and physical sciences. Examine key concepts including Margolis' motivational theorem, arithmetic lattices in discrete subgroups, and thin subgroups in the sense of Sarnak. Follow the strategic approach to constructing examples, and understand important theorems by Greenberg, Leininger-Long-Read, Koberda, and Borel. Investigate the action of L on Q=Tau/H, and explore the use of harmonic maps in the proposed strategy. This talk, part of a two-week workshop on differentiable and homogeneous dynamics, offers a deep dive into cutting-edge research in these interconnected fields.

Syllabus

Commensurators of thin Subgroups
Motivational theorem due to Margolis
Does characterized arithmetic lattices in discrete subgroups
Let HG be a thin subgroup this is in the sense of sernek
Illustration of strategy
First proposition Greenberg
Theorem Leininger-Long-Read
Proposition reduces shalom's question looking at H thin n with limit h is equal to all of the boundary
How to construct examples of such h?
Theorem Thomas Koberda
Strategy
Theorem Borel
Trick
Want to look at: Action of l on Q = tou/h
Observation
Proposition
Strategy Hormonic maps
Observation