Stationary Random Subgroups

Project Information

SRS

Grant agreement ID: 101052954

DOI

10.3030/101052954

EC signature date 27 April 2022

Start date 1 May 2022

End date 30 April 2028

Funded under

• European Research Council (ERC)

Total cost

€ 2 149 023,00

EU contribution

€ 2 149 023,00

Coordinated by

WEIZMANN INSTITUTE OF SCIENCE
Israel

Periodic Reporting for period 2 - SRS (Stationary Random Subgroups)

Reporting period: 2025-07-01 to 2028-04-30

Summary of the context and overall objectives of the project

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 realized 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. M. Fraczyk and I proved, using SRS, the following conjecture of Margulis: Let G be a higher rank simple Lie group and D a discrete subgroup. Then the orbifold D\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 D\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.

Work performed from the beginning of the project to the end of the period covered by the report and main results achieved so far

The main achievement of the project is a solution of the long standing Margulis injectivity radius conjecture in complete generality. The main theorem is that higher rank locally symmetric manifolds have bounded injectivity radius.
This is a very strong generalization of the celebrated Margulis normal subgroup theorem which was done in two separate works.

The first was a joint work of the PI with M. Fraczyk where the conjecture has been solved in the case where the ambient Lie group G has Kazhdan's property (T). When G has property (T) we have shown a stronger result, namely that a random walk on a manifold of infinite volume eventually escapes from any region of bounded injectivity radius. The proof is probabilistic in nature and this work has been quite influential since it demonstrated the power of the theory of stationary random subgroups. The theory of invariant random subgroups has been very popular in the last 15 years. However that theory has limited power and was not enough in order to tackle the Margulis conjecture. This lead us to consider non-invariant random subgroups. Our development of the theory of stationary random subgroups has already proven successful beyond the scope of the original motivation and has lead other people to investigate and develop that theory as well.

The second work, which was done in collaboration with U. Bader and A. Levit, dealt with the case where the ambient group does not have property (T). As in the case of the classical normal subgroup theorem of Margulis, the lack of property (T) adds a difficulty to the problem and requires to establish a certain new spectral gap result. This leads us to deal with random subgroups which are not even stationary. This was a significant technical challenge.

Both of these works can be considered breakthroughs. The first work, which was started before the duration of the project, has started a new field, namely the study of discrete stationary subgroups. The second work has established a new spectral gap which can serve in some situations as an alternative of Kazhdan's property (T).

Progress beyond the state of the art and expected potential impact (including the socio-economic impact and the wider societal implications of the project so far)

The main breakthrough of this project has been a new spectral gap theorem which allowed us to prove the general case of the infinite injectivity radius conjecture. The most difficult case is when the ambient group G is a product of rank one Lie groups without (T), for instance G=SL(2,R)xSL(2,R).
We established a general spectral gap theorem for actions of products of groups which may replace Kazhdan’s property (T) in various situations. This allowed us to prove that a confined subgroup of an irreducible lattice in a higher rank semisimple Lie group is of finite index. This significantly strengthens the classical normal subgroup theorem of Margulis. We further show that any confined discrete subgroup of a higher rank semisimple Lie group satisfying a certain irreducibility condition is an irreducible lattice. This implies a variant of the Stuck–-Zimmer conjecture under a strong irreducibility assumption of the action.