Zimmer’s conjecture for actions of $$\mathrm {SL}(m,\pmb {\mathbb {Z}})$$

Property FW, differentiable structures and smoothability of singular actions

“…Proof of Corollary 3.7. Both facts follow by combining the main results in [7,8] with Theorem 3.1, applied to the groupoids G = D 2 and G = D 2 Leb , respectively.…”

“…In particular, it remains a well-known open problem whether Zimmer's conjecture holds for all action by homeomorphisms. In combination with the aforementioned results of Brown, Fisher and Hurtado [7,8], Theorem 1.1 yields the following for countably singular actions. Using a more general version of Theorem 1.1 (namely Theorem 3.1), we also establish an analogous statement in the case of actions preserving a volume form, see Corollary 3.7.…”

“…Here we have simplified the statement in order to avoid introducing too much cumbersome notation. The interested reader will be able to improve the above statement from the results appearing in [6][7][8]10]. Note also that Theorem 3.6 gives a stronger result for actions by elements with finitely many singularities.…”

“…A central conjecture in Zimmer's program states that a lattice in a higher rank simple Lie group has only (virtually) trivial actions on closed manifolds of dimension < d, where d is an explicit constant depending on the ambient Lie group (bounded below by its real rank). The conjecture has been (partially) solved recently with the breakthrough work of Brown, Fisher and Hurtado [6][7][8] for a class of actions by C 2 -diffeomorphisms.…”

Book Review: Group actions in ergodic theory, geometry, and topology: Selected papers

“…Despite all of the advances, strategies for resolving the general form of Zimmer's conjecture remained elusive at the time of Fisher's survey. The situation has changed dramatically, however, with a recent breakthrough of Brown, Fisher, and Hurtado, who proved among more general statements that, for n > 2, every smooth action of a cocompact lattice in SL(n, R) on a compact manifold of dimension less than n − 1 factors through a finite quotient [2,3]. This achievement at the heart of the Zimmer program has given occasion to the additional text that Fisher has prepared as an afterward to the volume, a highly engaging personal narrative chronicling the pollination, percolation, and cross-fertilization of ideas that led to the results.…”