Spring Central Sectional Sampler

The Zimmer program refers to a number of questions and conjectures posed by Robert Zimmer in the 1980s concerning smooth actions of lattices in higher-rank Lie groups. We report on some recent progress by the author and collaborators. Lattices in Higher-Rank Lie Groups The primary objects inmy talk are lattices Γ in higher-rank simple Lie groups G. The simplest example of such a lattice is Γ = SL(n, Z), the group of n × n integral matrices with determinant one. The group Γ = SL(n, Z) is a discrete subgroup of G = SL(n,R), the group of n × n invertible matrices of determinant one. It is well known that the coset space G/Γ admits a finite-volume form which is invariant under the (left) action of G by translations on G/Γ. In general, a lattice in a Lie group G is a discrete subgroup Γ ⊂ G such that the coset space G/Γ has a finite G-invariant volume. A lattice is cocompact if the coset space G/Γ is compact. The primary example of a lattice subgroup, SL(n, Z) in SL(n,R), is not cocompact; there do, however, exist cocompact lattices in SL(n,R). The rank of the group G = SL(n,R) is n− 1; this is the dimension of the subgroup A = {diag(et1 , et2 ,... , etn)} of diagonal matrices with positive entries. (Note that since we impose that the determinant is 1, we have t1+t2+⋯+tn = 0.) When n ≥ 3, the rank of G = SL(n,R) is at least 2, and we say that G is higher rank. Rigidity of Linear Representations Given a discrete group Γ, a linear representation of Γ is a homomorphism π∶ Γ → GL(d,R) from Γ to a linear group GL(d,R). When Γ is a lattice in a higher-rank simple Lie group, it is well known that linear representations π∶ Γ → GL(d,R) exhibit a number of strong rigidity properties. Early rigidity results include the local rigidity results due to Selberg and Weil and the global rigidity results due to Mostow, Margulis, and Prasad. The strongest rigidity results for linear representations are the superrigidity and arithmeticity theorems of Margulis. Margulis’s superrigidity theorem roughly states that any linear representation π∶ Γ → GL(d,R) of a lattice Γ in a higher-rank simple Lie group G extends to a representation ?̃?∶ G → GL(d,R) “up to a compact error.” Aaron Brown is assistant professor of mathematics at the University of Chicago. His email address is [email protected]. For permission to reprint this article, please contact: [email protected]. DOI: http://dx.doi.org/10.1090/noti1643 In particular, since representations ?̃?∶ G → GL(d,R) are fully classified, the superrigidity theorem essentially classifies all linear representations of lattices in higher-rank Lie groups. As a motivating result for my talk, we have the following corollary of Margulis’s superrigidity theorem: Corollary. For n ≥ 3 and d < n, let Γ be a lattice in SL(n,R). Then the image of any linear representation