Amenable Invariant Random Subgroups

We show that an amenable Invariant Random Subgroup of a locally compact second countable group lives in the amenable radical. This answers a question raised in the introduction of the paper "Kesten's Theorem for Invariant Random Subgroup" by Abert, Glasner and Virag. We also consider, in the opposite direction, property (T), and prove a similar statement for this property. The Appendix by Phillip Wesolek proves that the set of amenable subgroups is a Borel subset in the Chabauty topology.Comment: We added an Appendix by Phillip Wesole

Almost algebraic actions of algebraic groups and applications to algebraic representations

Let G be an algebraic group over a complete separable valued field k. We discuss the dynamics of the G-action on spaces of probability measures on algebraic G-varieties. We show that the stabilizers of measures are almost algebraic and the orbits are separated by open invariant sets. We discuss various applications, including existence results for algebraic representations of amenable ergodic actions. The latter provides an essential technical step in the recent generalization of Margulis-Zimmer super-rigidity phenomenon due to Bader and Furman.Comment: Correction of a small mistake in Proposition 5.

Boundary maps and maximal representations on infinite dimensional Hermitian symmetric spaces

We define a Toledo number for actions of surface groups and complex hyperbolic lattices on infinite dimensional Hermitian symmetric spaces, which allows us to define maximal representations. When the target is not of tube type we show that there cannot be Zariski-dense maximal representations, and whenever the existence of a boundary map can be guaranteed, the representation preserves a finite dimensional totally geodesic subspace on which the action is maximal. In the opposite direction we construct examples of geometrically dense maximal representation in the infinite dimensional Hermitian symmetric space of tube type and finite rank. Our approach is based on the study of boundary maps, that we are able to construct in low ranks or under some suitable Zariski-density assumption, circumventing the lack of local compactness in the infinite dimensional setting.Comment: Comments are welcome! The maximality assumption was unfortunately missing in Theorem 1.1 and 1.4 of the first versio

Liquid ilmenite or liquidus ilmenite: a comment on the nature of ilmenite vein deposits

The interstitial character of ilmenite in common gabbroidic rocks has led to the idea that the last drop of liquid to crytallize in a rock could be pure ilmenite. When injected into enclosing rocks this liquid could give rise to ilmenite-pure veins. Fractional crystallization or immiscibility cannot produce a pure ilmenite melt. Actually, ilmenite is a liquidus mineral, which is the first or among the first mafic minerals to crystallize in Fe-Ti rich magmas. This is clearly shown by the sequence of crystallization in the Bjerkreim-Sokndal layered intrusion, by quantitative modelling of the liquid line of descent of jotunitic (hypersthene monzodioritic) magmas, and by experimental work. The interstitial character of ilmenite is therefore acquired subsolidus. Deformation can enhance the process. The synemplacement deformation linked to the diapiric uprise of anorthosite massifs is a suitable environment for the formation of layers of Cr- and Mg-rich ilmenite and their subsequent deformation and mobilization in discordant veins of pure ilmenite

Peculiar characteristics of the Bjerkreim-Sokndal intrusion (Rogaland anorthosite province, S. Norway) and implications for the layering formation

Version préliminair