SPRING CENTRAL SECTIONAL SAMPLER From left to right: Aaron Brown, Tullia Dymarz, and June Huh. In this sampler, the speakers above have kindly provided introductions to their Invited Addresses for the upcoming AMS Spring Central Sectional Meeting. March 17–18, 2018 (Saturday–Sunday) The Ohio State University Columbus, OH Recent Progress in the Zimmer Program by Aaron Brown (University of Chicago) page 308 Bi-Lipschitz Equivalence of Groups by Tullia Dymarz (University of Wisconsin— Madison) page 310 The Correlation Constant of a Field by June Huh (Institute for Advanced Study) page 311 For permission to reprint this article, please contact: [email protected]. DOI: http://dx.doi.org/10.1090/noti1671 SPRING CENTRAL SECTIONAL SAMPLER Aaron Brown In particular, since representations ̃휋∶ 퐺 → GL(푑, ℝ) are fully classified, the superrigidity theorem essentially clas- sifies all linear representations of lattices in higher-rank Recent Progress in the Zimmer Program Lie groups. As a motivating result for my talk, we have the following corollary of Margulis’s superrigidity theorem: ABSTRACT. The Zimmer program refers to a number Corollary. For 푛 ≥ 3 and 푑 < 푛, let Γ be a lattice in of questions and conjectures posed by Robert Zimmer SL(푛, ℝ). Then the image of any linear representation in the 1980s concerning smooth actions of lattices in higher-rank Lie groups. We report on some recent 휋∶ Γ → GL(푑, ℝ) progress by the author and collaborators. is a finite group. Actions on Manifolds and the Zimmer Program Lattices in Higher-Rank Lie Groups Given a compact manifold 푀, let Diff(푀) denote the The primary objects in my talk are lattices Γ in higher-rank group of smooth diffeomorphism of 푀. Given a fixed simple Lie groups 퐺. volume form on 푀, let Diffvol(푀) be the group of volume- The simplest example of such a lattice is Γ = SL(푛, ℤ), preserving diffeomorphisms of 푀.A smooth action of the group of 푛 × 푛 integral matrices with determinant Γ on 푀 (or a “nonlinear representation”) is simply a one. The group Γ = SL(푛, ℤ) is a discrete subgroup of homomorphism 퐺 = SL(푛, ℝ), the group of 푛 × 푛 invertible matrices of 훼∶ Γ → Diff(푀). determinant one. It is well known that the coset space Consider Γ = SL(푛, ℤ). We describe two prototype ac- 퐺/Γ admits a finite-volume form which is invariant under tions of Γ on low-dimensional compact manifolds; both the (left) action of 퐺 by translations on 퐺/Γ. In general, examples are derived from the natural action of SL(푛, ℤ) a lattice in a Lie group 퐺 is a discrete subgroup Γ ⊂ 퐺 by linear transformations on ℝ푛. First, note that matrix such that the coset space 퐺/Γ has a finite 퐺-invariant multiplication preserves rays in ℝ푛. Thus, viewing the volume. A lattice is cocompact if the coset space 퐺/Γ sphere 푆푛−1 as a parametrization of all rays in ℝ푛, we is compact. The primary example of a lattice subgroup, obtain a natural action of Γ = SL(푛, ℤ) on 푆푛−1. Concretely, SL(푛, ℤ) in SL(푛, ℝ), is not cocompact; there do, however, considering 푆푛−1 as the set of unit vectors in ℝ푛−1, we exist cocompact lattices in SL(푛, ℝ). 푛−1 The rank of the group 퐺 = SL(푛, ℝ) is 푛 − 1; this is the define 훼∶ Γ → Diff(푆 ) by 훾 ⋅ 푥 dimension of the subgroup 훼(훾)(푥) = . ‖훾 ⋅ 푥‖ 퐴 = {diag(푒푡1 , 푒푡2 , … , 푒푡푛 )} This action does not preserve any volume form on 푆푛−1. of diagonal matrices with positive entries. (Note that For the second action, since for every 훾 ∈ SL(푛, ℤ) the since we impose that the determinant is 1, we have coefficients of 훾 are integral and det 훾 = 1, the action of 푡 + 푡 + ⋯ + 푡 = 0.) When 푛 ≥ 3, the rank of 퐺 = SL(푛, ℝ) 1 2 푛 Γ = SL(푛, ℤ) on ℝ푛 by linear transformations preserves the is at least 2, and we say that 퐺 is higher rank. subgroup ℤ푛 ⊂ ℝ푛; this induces an action of Γ = SL(푛, ℤ) on the 푛-dimensional torus 핋푛 = ℝ푛/ℤ푛. Concretely, we Rigidity of Linear Representations define 훼∶ Γ → Diff(핋푛) by Given a discrete group Γ, a linear representation of Γ is 푛 푛 a homomorphism 휋∶ Γ → GL(푑, ℝ) from Γ to a linear 훼(훾)(푥 + ℤ ) = 훾푥 + ℤ . group GL(푑, ℝ). When Γ is a lattice in a higher-rank simple This action of SL(푛, ℤ) preserves the natural Lebesgue Lie group, it is well known that linear representations volume form on 핋푛. 휋∶ Γ → GL(푑, ℝ) exhibit a number of strong rigidity The Zimmer program refers to a number of questions properties. Early rigidity results include the local rigidity and conjectures posed by Robert Zimmer in the 1980s results due to Selberg and Weil and the global rigidity concerning smooth actions of lattices in higher-rank results due to Mostow, Margulis, and Prasad. Lie groups. Roughly, these questions and conjectures The strongest rigidity results for linear representations aim to establish analogues of known rigidity results are the superrigidity and arithmeticity theorems of Mar- for linear representations in the “nonlinear” setting of gulis. Margulis’s superrigidity theorem roughly states that smooth actions on manifolds. In particular, one may any linear representation 휋∶ Γ → GL(푑, ℝ) of a lattice Γ expect that all smooth actions of higher-rank lattices Γ in a higher-rank simple Lie group 퐺 extends to a rep- are constructed from standard algebraic actions (such resentation ̃휋∶ 퐺 → GL(푑, ℝ) “up to a compact error.” as the actions described in the previous paragraph) and thus, in some sense, can be completely classified. Aaron Brown is assistant professor of mathematics at the Univer- A first step towards classifying all actions is Zimmer’s sity of Chicago. His email address is [email protected]. conjecture, which asserts that no nontrivial1 actions exist For permission to reprint this article, please contact: [email protected]. 1An action 훼∶ Γ → Diff(푀) is trivial if the image 훼(훾) ⊂ Diff(푀) DOI: http://dx.doi.org/10.1090/noti1643 is a finite group; otherwise, it is said to be nontrivial. 308 Notices of the AMS Volume 65, Number 3 SPRING CENTRAL SECTIONAL SAMPLER on closed manifolds whose dimension is less than the showing local and global rigidity of volume-preserving dimension of any manifold appearing in an algebraic Anosov actions of higher-rank lattices on tori. action. For instance, when Γ = SL(푛, ℤ), the action on the In recent work with Rodriguez Hertz and Wang, for sphere 푆푛−1 is smallest dimensional algebraic action and 푛 ≥ 3 we study actions of Γ = SL(푛, ℤ) (and more general the action on the torus 핋푛 is smallest dimensional volume- higher-rank lattices) on tori 핋푑 (and nilmanifolds). Under preserving algebraic action. In particular, for lattices in certain dynamical hypotheses on the induced action on SL(푛, ℝ), Zimmer’s conjecture asserts the following. homology and a mild lifting condition, we show that such an action is, up to a continuous surjective (but Zimmer’s Conjecture. For 푛 ≥ 3, let Γ ⊂ SL(푛, ℝ) be a possibly noninvertible) change of coordinates, given by lattice and let 푀 be a closed manifold. an algebraic action. For Anosov actions, the change of coordinates is then shown to be a smooth diffeomorphism. (1) If dim(푀) < 푛 − 1, then any homomorphism Unlike earlier approaches, our method does not require 훼∶ Γ → Diff푟(푀) that the action preserve a volume (or any probability measure). has finite image. In another recent and ongoing work with Rodriguez Hertz and Wang, for 푛 ≥ 3 we study actions of Γ = SL(푛, ℤ) (2) If dim(푀) < 푛 and 훼 preserves a smooth volume on (푛 − 1)-dimensional manifolds. For such actions, we form, then any homomorphism showed that all nontrivial actions are measurably equiva- 푟 훼∶ Γ → Diffvol(푀) lent to (a finite measurable cover of) the standard action has finite image. on the sphere; in ongoing work, we aim to show all such actions are equivalent to the natural action on the In particular, Zimmer’s conjecture provides an analogue sphere (or projective space) up to a smooth change of in the context of smooth actions of the finiteness of all coordinates. linear representations given by the corollary to Margulis’s The proofs of the above results combine tools and re- superrigidity theorem above. Prior work towards Zim- sults from many areas of mathematics, including smooth mer’s conjecture primarily focused on actions on the ergodic theory, homogeneous dynamics, Lie theory, repre- circle and on surfaces, including work by Witte, Burger sentation theory, and operator algebras. In my talk I will and Monod, Ghys, Polterovich, and Franks and Handel. present a number of motivations for the Zimmer program, outline the new results discussed above, and indicate how Recent Progress in the Zimmer Program tools and ideas from various areas of mathematics are My talk will focus on recent work on Zimmer’s conjecture combined to establish these results. and the larger Zimmer program due to David Fisher, Sebastian Hurtado, Federico Rodriguez Hertz, Zhiren Wang, and me. ABOUT THE AUTHOR The first main result is the verification of Zimmer’s conjecture for actions of certain lattices in SL(푛, ℝ). Aaron Brown’s research is in Namely, when Γ ⊂ SL(푛, ℝ) is either smooth dynamics, ergodic theory, and group actions on manifolds. • a cocompact lattice in SL(푛, ℝ) or • a finite-index subgroup of SL(푛, ℤ), Fisher, Hurtado, and I verified that Zimmer’s conjecture holds.