Strong property (T) for higher-rank simple Lie groups: Figure 1.

Abstract

Abstract. We prove that connected higher rank simple Lie groups have Lafforgue's strong property (T) with respect to a certain class of Banach spaces E 10 containing many classical superreflexive spaces and some non-reflexive spaces as well. This generalizes the result of Lafforgue asserting that SL(3, R) has strong property (T) with respect to Hilbert spaces and the more recent result of the second named author asserting that SL(3, R) has strong property (T) with respect to a certain larger class of Banach spaces. For the generalization to higher rank groups, it is sufficient to prove strong property (T) for Sp(2, R) and its universal covering group. As consequences of our main result, it follows that for X ∈ E 10 , connected higher rank simple Lie groups and their lattices have property (F X ) of Bader, Furman, Gelander and Monod, and that the expanders contructed from a lattice in a connected higher rank simple Lie group do not admit a coarse embedding into X.

On strong property (T) and fixed point properties for Lie groups

“…We refer to the beginning of Section 4 for further comparison between Theorem A and the work in [15]. (iv) As in [29] and [14], the conclusion of Theorem A also holds with N > max(8, 3 1−2β − 3) if X is a complex interpolation space between a Banach space satisfying (1) and an arbitrary Banach space. It is unknown whether there exists a Banach space satisfying (1) for some β < 1 2 (or more generally a space of type > 1) which is not a complex interpolation space between a space satisfying (1) for β = 10 −10 and an arbitrary Banach space.…”

“…This is still open. The first steps towards such a result were provided in [29] and [14], the main results of which imply that every connected simple Lie group with real rank at least 2 has strong property (T) with respect to the Banach spaces satisfying (1) for β < 1 10 . Theorem A is another step towards a real analogue of Lafforgue's and Liao's results.…”

The p-approximation property for simple Lie groups with finite center

“…This is achieved by considering two different families of operators that will play the role of (T δ ), namely (15) and (16). These operators were defined in [17]. From now on we fix p ∈ [2, ∞), G = Sp(2, R) and K as in (5).…”

“…This allows, by interpolation, to obtain the same kind of results for pcompletely bounded multipliers. These averaging operators were already considered in [18], [19] and [17]. Also, we don't make use of the Krein-Smulian theorem.…”

Superexpanders from group actions on compact manifolds

“…In particular, this implies that these expanders are superexpanders, since the class of Banach spaces with nontrivial type is known to be larger than the class of uniformly convex spaces. More examples of groups with strong property (T) or other Banach space versions of property (T) relative to various classes of Banach spaces were provided in [11][12][13]16,22,25]. Other examples of superexpanders, obtained by means of a combinatorial construction, were provided in the groundbreaking work of Mendel and Naor [20].…”