The first two lectures will serve to recall the needed background in group theory and operator algebras. Since time is limited, these lectures will be fast-paced overviews. It is expected that the participants familiarize themselves beforehand with the concepts appearing in Lectures 1 and 2.
Recall the definition and basic properties of higher rank lattices (definition, a uniform and a nonuniform example in SLn, property (T), finiteness properties). Expose the statements of local rigidity, (strong) rigidity and superrigidity. State Margulis' normal subgroup theorem and explain the original proof strategy (in particular, mention Margulis' factor theorem) [Z].
State the Stuck-Zimmer theorem on ergodic actions of a higher rank Lie group G, and of a higher rank lattice Γ [SZ]. Deduce from it the classification of IRS in G, and in turn Margulis' normal subgroup theorem (see [G]). Use this (and other examples at will) to motivate dynamical methods in the study of discrete groups.
If time permits, give more details on the Chabauty space Sub(G) and introduce URS.
Recall the construction of group C*- and vN-algebras, their basic properties, and the GNS construction in detail [BdlH, sections 1.B]. Define characters, positive functions, the different kinds of states. Explain how these objects appear naturally when a group is acting on various objects (see for instance [BdlH, example 1.B.7, chapter 13, and proposition 15.F.3–4]).
State Bekka's theorem on the operator-algebraic superrigidity of SLn(Z). Introduce operator-algebraic superrigidity, and show it is equivalent to character rigidity of the group.
Optional: Draw a brief comparison between the topological and measurable worlds. Mention Connes' rigidty conjecture. Comment on the absence of rigidity for amenable groups.
State Peterson's character rigidity theorem, and show it implies the Stuck–Zimmer theorem.
State theorems A (representation rigidity), B (URS rigidity / topological analogue of S–Z), C (classification of stationary characters), and D (structure of stationary actions) from [H1] (see also [H2] and [BBHB]). To do so, briefly introduce Furstenberg measures (definition 1.4) while defering any details to lectures 4 and 5.
Show how theorem A implies Margulis' normal subgroup theorem, the Stuck–Zimmer theorem, and Peterson's character rigidity theorem [H1, proposition 1.1]. Deduce theorem A from C, and theorem B from D. Comment how, in both arguments, Furstenberg measures basically allow to perform an "averaging trick".
Optional: Some of the above implications are redundant and may be skipped if time is insufficient. Mention Connes' rigidty conjecture if it hasn't been done in Lecture 2.
Define group boundaries, and the universal (sometimes called "maximal") boundary of a group. Prove its existence and universal property. Describe the universal boundary when G has cocompact amenable subgroups [O, proposition 10].
Construct the Poisson boundary of a group and the Poisson transform; discuss their basic properties. Give an implicit definition of the Poisson boundary in terms of measure G-spaces.
Give a description of the universal and Poisson boundaries of a simple real Lie group G. Idem for an amenable group. State Furstenberg's theorem on the existence of Furstenberg measures (in other words, identify the Poisson boundary of a lattice Γ with that of G). Explain why the universal boundary of the lattice Γ is very different [O, proposition 8].
If time permits: Sketch the proof of the existence of Furstenberg measures. Mention the connection between C*-simplicity and the existence of a topologically free G-boundary [O, theorem 15].
Recall Furstenberg measures and stationarity. Prove theorem C using theorem D. Construct the induced group vN-algebras associated to an action of Γ, and show that stationary characters can be induced as well [H1, theorem 2.4]. Conclude that one could attempt to prove theorem D by proving an analogous structure theorem for stationary actions of G this time: the non-commutative Nevo–Zimmer theorem.
If time permits: Introduce the formalism of boundary structures (see [H2, section 3]).
State the original Nevo–Zimmer theorem, and its non-commutative analogue, theorem E in [H1]. Deduce theorem D from theorem E (see [BH, section 6]), as promised in the previous lecture. Deduce from theorem E a non-commutative analogue of Margulis' factor theorem [H2, corollary F]. Lay out the strategy of the proof of theorem E (see namely [H2, section 3.3]).
Can be restricted to the specific case Γ = PGL3(Z) in G = PGL3(R).
Can be restricted to the specific case Γ = PGL3(Z) in G = PGL3(R).