Let \(\Gamma\) be a lattice in a non-compact, simple Lie group \(G\) of rank \(rk(G)\ge 2\). Let \(n(G)\) be the minimal dimension of a real representation of \(G\). Zimmer’s conjecture asserts that actions of \(\Gamma\) on a closed manifold \(M\) with \(\dim(M)<n(G)\) should factor over the action of a finite group. Results in this direction had so far either involved assumptions on preservation of a probability measure or on the regularity of the action, or had only concerned actions on the circle or on surfaces. The paper under review proves the existence of an invariant probability measure for Holder actions on manifolds of “moderately low dimension”.
Denote by \(r(G)\) the minimal resonant codimension of \(G\). This is \(n\) for Lie groups of type \(A_n\), \(2n-1\) for type \(B_n\) and \(C_n\), and \(2n-2\) for type \(D_n\) with \(n\ge 4\). The authors prove that there is an invariant probability measure for every Holder action of \(\Gamma\) on a closed manifold \(M\) with \(\dim(M)<r(G)\). In the case \(\dim(M)=r(G)\) they prove that either there is an invariant probability measure or there is a quasi-invariant probability measure such that the action is measurably conjugate to the standard action of \(\Gamma\) on \(G/Q\) for some maximal parabolic \(Q\subset G\).
More generally, the authors define a number \(m(G)\), which is the minimal value of the resonant codimension over certain parabolic subgroups. Its value is \(2n-1\) for Lie groups of type \(A_n\), \(4n-4\) for type \(B_n\) and \(C_n\), and \(4n-6\) for \(D_n\) with \(n\ge 5\). Then the authors prove that for an action with \(\dim(M)\le m(G)\) there is a quasi-invariant probability measure such that the action is a relatively measure-preserving extension of the standard action on \(G/Q\) for some maximal parabolic \(Q\subset G\).
As an application, the authors can complete the results of J. Franks and M. Handel [Duke Math. J. 131, No. 3, 441–468 (2006; Zbl 1088.37009)] to prove that for a non-uniform lattice in \(\mathrm{SL}(n,{\mathbb R}), n\ge 4\), any Holder action on a surface of genus at least \(1\) factors over the action of a finite group.
The methods used in the paper under review have shown to be important for the proof of Zimmer’s conjecture for cocompact lattices in \(\mathrm{SL}(n,\mathbb R)\) by A. Brown et al. [Ann. Math. (2) 196, No. 3, 891–940 (2022; Zbl 1508.22013)]. There is first the construction of a “suspension space” \(M^\alpha\), which is an \(M\)-bundle over \(\Gamma\backslash G\) and which comes with a Lie group action by \(G\) rather than the original \(\Gamma\)-action \(\alpha\) on \(M\). Further the proofs of the main theorems invoke the principle “non-resonance implies invariance” (see Proposition 5.1), which is also of importance for the proof of Zimmer’s conjecture.