The Weak Pinsker Property (WPP) asks whether an ergodic probability-measure-preserving action can always be split, up to measure conjugacy, as a direct product of a Bernoulli shift and an action of arbitrarily small entropy. Austin proved this for every ergodic transformation and, more generally, for ergodic actions of amenable groups, while Bowen constructed the first counterexample: an ergodic action of a non-abelian free group of sufficiently large rank that fails the WPP with respect to a suitable sofic approximation, detected by newly introduced sofic homology invariants. Later work of Austin–Bowen–Shriver extended the counterexample phenomenon to certain free products of finite cyclic groups and produced a positive-sofic-entropy action with no nontrivial direct Bernoulli factors at all. The question addressed here is the next question Bowen posed, still open: does every non-amenable sofic group admit an ergodic action (with respect to some sofic approximation) that fails the Weak Pinsker Property?

Background

A probability-measure-preserving (pmp) action of a countable group Γ\Gamma is a homomorphism from Γ\Gamma into the measure-preserving transformations of a standard probability space (X,μ)(X,\mu). The basic building blocks are Bernoulli shifts: for a base probability space (k,κ)(\mathsf{k},\kappa), the shift acts on the product space kΓ\mathsf{k}^{\Gamma} (configurations (xg)g∈Γ(x_g)_{g\in\Gamma} with independent, identically distributed coordinates) by permuting indices, x↦(xg−1f)fx\mapsto (x_{g^{-1}f})_{f}. Two actions are measurably conjugate (isomorphic) if a measure-space isomorphism intertwines them. An action is ergodic if every invariant set has measure 0 or 1. For actions of Z\mathbb{Z} (and, with more work, of amenable groups — those admitting an invariant finitely-additive mean, equivalently a Følner sequence), entropy is the classical Kolmogorov–Sinai entropy rate. Pinsker conjectured in 1960 that every ergodic transformation is a direct product of a zero-entropy transformation and a transformation all of whose nontrivial factors have positive entropy; Ornstein disproved this. In the 1970s Thouvenot introduced the Weak Pinsker Property (WPP) as a residual form of Pinsker's question: a transformation TT has the WPP if for every ϵ>0\epsilon>0 it is measurably conjugate to a direct product S×US\times U where SS has entropy <ϵ<\epsilon and UU is a Bernoulli shift, and he asked whether every ergodic transformation has it. In breakthrough work, Austin proved that every ergodic transformation does have the WPP, and the analogous statement holds for ergodic actions of every amenable group. Outside amenability the right notion of entropy is subtle. A group Γ\Gamma is sofic if it admits a sofic approximation Σ=(σn)\Sigma=(\sigma_n), a sequence of almost-actions on finite sets: homomorphisms-up-to-small-error σn:Γ→Sym(Vn)\sigma_n:\Gamma\to\mathrm{Sym}(V_n) into finite symmetric groups that respect multiplication and move almost every point under every non-identity element. All amenable groups and all residually finite groups are sofic; whether all countable groups are sofic is open. For a shift-invariant measure μ\mu on a finite alphabet kΓ\mathsf{k}^{\Gamma}, the model space Ω(O,σn)\Omega(\mathcal{O},\sigma_n) is the set of labelings of VnV_n whose local patterns realize an open neighborhood O\mathcal{O} of μ\mu, and the sofic entropy (or Σ\Sigma-entropy) hΣ(μ)h_\Sigma(\mu) is the exponential growth rate of the cardinalities of these model spaces; it is a measure-conjugacy invariant that depends a priori on Σ\Sigma and coincides with classical entropy for amenable groups. Rokhlin entropy, defined by Seward as the infimum of the Shannon entropy of a generating partition, upper bounds Σ\Sigma-entropy and is defined for all countable groups. Bowen defined the WPP with respect to Σ\Sigma for sofic-group actions verbatim: for every ϵ>0\epsilon>0 the action is isomorphic to a direct product b×c\mathsf{b}\times\mathsf{c} with b\mathsf{b} a Bernoulli shift and hΣ(c)<ϵh_\Sigma(\mathsf{c})<\epsilon. He then proved that for some r0r_0, every free group Fr\mathbb{F}_r of rank r>r0r>r_0 admits a sofic approximation Σ\Sigma and an ergodic essentially free pmp action — one in which every non-identity group element moves almost every point of the underlying probability space — without the WPP with respect to Σ\Sigma (hence also without the a priori stronger WPP with respect to Rokhlin entropy). The counterexample is a weak* limit of hard-core models — Gibbs measures on independent sets, configurations with no two adjacent occupied vertices — on random regular graphs, and its analysis introduced new measure-conjugacy invariants: the dd-dimensional sofic homology groups of the model spaces and their exponential and polynomial growth rates. The key mechanism is that any action with the WPP has subexponential growth of 0-dimensional sofic homology, while the hard-core limit has exponential 0-dimensional growth. Subsequently Austin–Bowen–Shriver, using parity-check subshifts (algebraic subshifts defined by local parity constraints, analyzed through random low-density-parity-check codes) over the free products Γd,k=(Z/kZ)∗d\Gamma_{d,k}=(\mathbb{Z}/k\mathbb{Z})^{*d} with k>d≥3k>d\ge 3, exhibited systems with completely positive sofic entropy (with respect to `most' sofic approximations) and no nontrivial direct Bernoulli factors whatsoever, whose model spaces shatter into exponentially many well-separated clusters. These constructions, however, are tied to free groups and free products of finite groups, and the general question for arbitrary non-amenable sofic groups remains. Austin–Bowen–Shriver pose the same question for arbitrary non-amenable groups, noting that sofic entropy may be insufficient for that general formulation.

Problem Statement

For every non-amenable sofic group Γ\Gamma, does there exist a sofic approximation Σ=(σn)\Sigma=(\sigma_n) to Γ\Gamma and an ergodic probability-measure-preserving action of Γ\Gamma that does not have the Weak Pinsker Property with respect to Σ\Sigma — that is, an ergodic action for which it fails that, for every ϵ>0\epsilon>0, it is measurably conjugate to a direct product b×c\mathsf{b}\times\mathsf{c} of a Bernoulli shift b\mathsf{b} over Γ\Gamma and an action c\mathsf{c} with Σ\Sigma-entropy hΣ(c)<ϵh_\Sigma(\mathsf{c})<\epsilon? Bowen's paper establishes this for free groups of sufficiently large rank, and Austin–Bowen–Shriver extend it to free products (Z/kZ)∗d(\mathbb{Z}/k\mathbb{Z})^{*d} with k>d≥3k>d\ge 3; the question asks whether it holds for every non-amenable sofic group (for example, for property (T) sofic groups — groups all of whose unitary representations admit a uniform spectral gap, such as lattices in higher-rank semisimple Lie groups — or for torsion sofic groups). Since Rokhlin entropy upper bounds Σ\Sigma-entropy, an action failing the WPP with respect to some Σ\Sigma also fails the a priori stronger WPP with respect to Rokhlin entropy.

References

  1. 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; arXiv:1807.08191. https://arxiv.org/abs/1807.08191
  2. J.-P. Thouvenot, "On the stability of the weak Pinsker property", Israel Journal of Mathematics 27(2):150–162, 1977. DOI 10.1007/BF02761664. https://doi.org/10.1007/BF02761664
  3. 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; arXiv:1705.00302. https://arxiv.org/abs/1705.00302
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