For which acting groups does the weak Pinsker property hold universally — that is, every free ergodic probability-preserving action splits as a Bernoulli shift times a factor of arbitrarily small entropy? Austin's measure-concentration theorem established the property for all ergodic Z\mathbb{Z}-actions, all free ergodic actions of countable amenable groups, and all ergodic flows, while Bowen's sofic-homology invariants show it can fail for free groups of sufficiently large rank. The precise open core is: (1) whether every non-amenable sofic group admits an essentially free ergodic p.m.p. action failing the weak Pinsker property with respect to some sofic approximation — making amenability exactly the obstruction among sofic groups — and (2) whether vanishing zero-dimensional sofic homology characterizes the property. Both remain open, including for free groups of small rank below Bowen's threshold.

Background

A probability-preserving (p.m.p.) action Γ↷(X,μ)\Gamma \curvearrowright (X,\mu) of a countable group Γ\Gamma is a measure-preserving transformation of a standard probability space for each group element. The Kolmogorov–Sinai entropy of such a system measures its asymptotic randomness; for a single transformation (Z\mathbb{Z}-action) the Pinsker factor is the largest quotient system of zero entropy, and the weak Pinsker property (WPP), introduced by Thouvenot in 1977, asks that for every ε>0\varepsilon>0 the system be measurably isomorphic to a product of a Bernoulli shift (the shift on a product of independent identically distributed variables, the prototypical completely random system) and a system of entropy at most ε\varepsilon. Thouvenot asked whether every ergodic transformation has the WPP; Austin resolved this affirmatively using measure-concentration arguments, and his paper further extends the result in two directions: Theorem 15.1 covers every free ergodic p.m.p. action of an arbitrary countable amenable group (amenable means the group admits a finitely additive invariant probability measure; the reduction passes through relative orbit equivalence in the spirit of Connes–Feldman–Weiss, with Rudolph–Weiss relative entropy as the key tool), and Proposition 16.16 covers every ergodic flow (R\mathbb{R}-action) via a criterion of Fieldsteel from 1984.

Beyond amenability the picture changes. A countable group is sofic if it embeds into metric ultraproducts of finite symmetric groups — a class encompassing both amenable and residually finite groups, and the largest class for which a satisfactory entropy theory is currently known. Lewis Bowen extended entropy to p.m.p. actions of sofic groups; sofic entropy is computed with respect to a chosen sofic approximation Σ\Sigma (a sequence of almost-homomorphisms into finite symmetric groups), and it can depend on Σ\Sigma. Bowen further introduced sofic homology: for each model space of the action one obtains finite abelian groups whose ranks, summarized by Betti numbers such as bΣ,0b_{\Sigma,0}, yield conjugacy invariants of the action. Two of his results delimit the current frontier. First, having the WPP with respect to Σ\Sigma forces bΣ,0=0b_{\Sigma,0}=0 (essentially, the zeroth model-space homology grows at most subexponentially), so vanishing sofic homology is a necessary condition. Second, for free groups Fr\mathbb{F}_r of sufficiently large rank rr there are invariant measures on {0,1}Fr\{0,1\}^{\mathbb{F}_r} — obtained as limits of hard-core models (independent sets) on random regular graphs, via shattering of the independent-set space — with bΣ,0>0b_{\Sigma,0}>0, hence essentially free ergodic actions failing the WPP with respect to some sofic approximation. More recently, Austin–Bowen–Shriver exhibited, for the groups Γd,k=⟨s1,…,sd∣sik=e⟩\Gamma_{d,k}=\langle s_1,\dots,s_d \mid s_i^k=e\rangle with k>d≥3k>d\ge 3, algebraic dynamical systems built from LDPC codes (low-density parity-check codes) whose sofic entropy is completely positive and which admit no nontrivial Bernoulli direct factor at all, a strong negation of the WPP. The boundary between the universal positive results (amenable groups, flows) and these negative examples is wide open, and Bowen's paper states the boundary questions explicitly as open problems.

Problem Statement

Let Γ\Gamma be a countable sofic group and let Σ\Sigma be a sofic approximation. Say that an ergodic p.m.p. action Γ↷(X,μ)\Gamma\curvearrowright(X,\mu) has the weak Pinsker property with respect to Σ\Sigma if it is measurably conjugate to a product of a Bernoulli shift and an action whose entropy (in the sense of the given sofic entropy theory) is arbitrarily small; let bΣ,0b_{\Sigma,0} denote its zeroth sofic homology Betti number. Two linked questions, stated at the generality of the sources (Bowen's problems 8 and 9; Austin–Bowen–Shriver §11.1):

(1) The group boundary. Does every non-amenable sofic group Γ\Gamma admit a sofic approximation Σ\Sigma and an essentially free ergodic p.m.p. action that fails the weak Pinsker property with respect to Σ\Sigma? Equivalently: among sofic groups, is amenability precisely the class of groups all of whose essentially free ergodic p.m.p. actions have the weak Pinsker property? This is open even for the free groups Fr\mathbb{F}_r of ranks below Bowen's threshold, including F2\mathbb{F}_2.

(2) The homological characterization. If an ergodic p.m.p. action of a sofic group has vanishing zero-dimensional sofic homology (bΣ,0=0b_{\Sigma,0}=0), must it have the weak Pinsker property with respect to Σ\Sigma — i.e., does the known necessary condition become a characterization?

Both questions concern the same boundary of the weak Pinsker phenomenon: bΣ,0>0b_{\Sigma,0}>0 is currently the only known obstruction, so an affirmative answer to (2) would make the obstruction complete, while (1) asks how far beyond amenability the obstruction is realized.

References

  1. Tim Austin, "Measure concentration and the weak Pinsker property", Publications Mathématiques de l'IHÉS 128:1–119 (2018). DOI: 10.1007/s10240-018-0098-3, https://arxiv.org/abs/1705.00302
  2. Lewis Bowen, "Sofic homological invariants and the Weak Pinsker Property", American Journal of Mathematics 144(1):169–226 (2022). DOI: 10.1353/ajm.2022.0003, https://arxiv.org/abs/1807.08191
  3. Tim Austin, Lewis Bowen, Christopher Shriver, "Algebraic dynamical systems from LDPC codes satisfy a strong negation of the weak Pinsker property", Journal of the Institute of Mathematics of Jussieu 24:2317–2385 (2025). DOI: 10.1017/S147474802510100X, https://arxiv.org/abs/2312.17387
  4. Lewis Bowen, Robin Tucker-Drob, "Superrigidity, measure equivalence, and weak Pinsker entropy", Groups, Geometry, and Dynamics 16(1):247–286 (2022). DOI: 10.4171/ggd/647, https://arxiv.org/abs/1805.03552
  5. J.-P. Thouvenot, "On the stability of the weak Pinsker property", Israel Journal of Mathematics 27:150–162 (1977). DOI: 10.1007/BF02761664, https://doi.org/10.1007/BF02761664
  6. Adam Fieldsteel, "Stability of the weak Pinsker property for flows", Ergodic Theory and Dynamical Systems 4:381–390 (1984). DOI: 10.1017/S0143385700002522, https://doi.org/10.1017/S0143385700002522
  7. Séverin Benzoni, "Introduction to weak Pinsker filtrations" (2025). arXiv:2504.00681, https://arxiv.org/abs/2504.00681