Stabilizers for Ergodic Actions of Higher Rank Semisimple Groups

Arithmeticity of Totally Geodesic Lie Foliations with Locally Symmetric Leaves

“…We refer to [9] for the complete statement. We observe that the result in [9] is stated for foliations with a simply connected dense leaf, but the existence of one such leaf is guaranteed just from the presence of a dense one by the main result in [8].…”

Rank gradient, cost of groups and the rank versus Heegaard genus problem

“…Note that a deep result of Stuck and Zimmer [30] tells us that every faithful ergodic measure preserving action of a higher rank semisimple real lattice on a probability space is essentially free. In particular, every faithful boundary representation of such a lattice is essentially free.…”

Invariant random subgroups of linear groups

“…The term "IRS" was introduced in a pair of joint papers with Abért and Virág [AGV13,AGV14], however the paper of Stuck-Zimmer [SZ94] is quite commonly considered as the first paper on this subject. That paper provides a complete classification of IRS in a higher rank simple Lie group G, by showing that every ergodic IRS is supported on a single orbit (i.e.…”

“…This is out of reach even for lattices in higher rank simple Lie groups, where we do have a complete understanding of seemingly similar problems such as the classification of all quotient groups or of all finite dimensional representations. An outcome of the Stuck-Zimmer paper [SZ94] is that, in the presence of an invariant measure -an IRS -the situation changes dramatically. For a lattice in a simple Lie group almost every subgroup, with respect to any IRS, is either finite central or of finite index.…”