Project description
Study leverages stationary random subgroups to study manifolds
Invariant random subgroups have proven extremely useful to the study of lattices and their asymptotic invariants. However, restricting to invariant measures limits the scope of problems that can be investigated. The EU-funded SRS project will leverage stationary random subgroups, which are much more general and help analysing discrete subgroups of infinite co-volume groups, especially of thin subgroups of arithmetic groups. Researchers will seek to solve the variant of the Schoen–Yau conjecture postulated by Margulis, namely higher-rank, locally symmetric manifolds Λ\G/K of infinite volume are not Liouville. A positive answer would have many applications in the theory of discrete subgroups of Lie groups.
Objective
The notion of invariant random subgroups (IRS) has proven extremely useful during the last decade, particularly to the study of asymptotic invariants of lattices. However, the scope of problems that one can investigate when restricting to invariant measures (on the space of subgroups) is limited. It was recently realised that the notion of stationary random subgroups (SRS), which is much more general, is still extremely powerful and opens up new paths to attacking problems that previously seemed to be out of our reach.
Notably, the notion of stationary random subgroups has turned out to be a wonderful new tool in the analysis of discrete subgroups of infinite co-volume, and, in particular, thin subgroups of arithmetic groups. A few months ago M. Fraczyk and I proved, using SRS, the following conjecture of Margulis: Let G be a higher rank simple Lie group and Λ ⊂ G a discrete subgroup. Then the orbifold Λ\G/K has finite volume if and only if it has bounded injectivity radius. This is a far-reaching generalisation of the celebrated Normal Subgroup Theorem of Margulis, and while it is new even for subgroups of lattices, it is completely general.
One of the main problems we wish to solve is the variant of the Schoen–Yau Conjecture postulated by Margulis; namely, that higher rank, locally symmetric manifolds Λ\G/K of infinite volume are not Liouville. A positive answer would have many applications in the theory of discrete subgroups of Lie groups. Some exiting applications are possible using partial results.
Fields of science
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Keywords
Programme(s)
- HORIZON.1.1 - European Research Council (ERC) Main Programme
Topic(s)
Funding Scheme
HORIZON-ERC - HORIZON ERC GrantsHost institution
7610001 Rehovot
Israel