Weak containment of measure-preserving group actions

In this chapter we specialize some of the considerations of Chap. 5 to the case of unitary C 0-groups in a Hilbert space ℋ. The theory of unitary representations W(x) = e iA·x of ℝn is a very well understood classical subject and will not be presented here.

We prove a pointwise ergodic theorem for quasi-probability-measure-preserving (quasi-pmp) locally countable measurable graphs, equivalently, Schreier graphs of quasi-pmp actions of countable groups. For ergodic graphs, the theorem gives an increasing sequence of Borel subgraphs with finite connected components over which the averages of any $L^1$ function converges to its expectation. This implies that every (not necessarily pmp) locally countable ergodic Borel graph on a standard probability space contains an ergodic hyperfinite subgraph. A consequence of this is that every ergodic treeable equivalence relation has an ergodic hyperfinite free factor. The pmp case of the main theorem was first proven by R. Tucker-Drob using a deep result from probability theory. Our proof is different: it is self-contained and applies more generally to quasi-pmp graphs. Among other things, it involves introducing a graph invariant concerning asymptotic averages of functions and a method of tiling a large part of the space with finite sets with prescribed properties. The non-pmp setting additionally exploits a new quasi-order called visibility to analyze the interplay between the Radon--Nikodym cocycle and the graph structure, providing a sufficient condition for hyperfiniteness.

In this paper, we consider the structure of the $L^2$-spectrum of the sub-Laplacian on 2-step stratified Lie groups by using the theory of unitary irreducible representations and Hermit functions. We extend the results for Heisenberg group and H-type Lie group to more general 2-step stratified Lie groups without the Moore-Wolf condition.

Perturbation theory for groups and lattices

This article is part of an ongoing project aiming at the connections between causal structures on homogeneous spaces, Algebraic Quantum Field Theory, modular theory of operator algebras and unitary representations of Lie groups. In this article we concentrate on non-compactly causal symmetric spaces G/H. This class contains de Sitter space but also other spaces with invariant partial ordering. The central ingredient is an Euler element h in the Lie algebra of $${{\mathfrak {g}}}$$ . We define three different kinds of wedge domains depending on h and the causal structure on G/H. Our main result is that the connected component containing the base point eH of these seemingly different domains all agree. Furthermore we discuss the connectedness of those wedge domains. We show that each of these spaces has a natural extension to a non-compactly causal symmetric space of the form $$G_{{\mathbb {C}}}/G^c$$ where $$G^c$$ is a certain real form of the complexification $$G_{{\mathbb {C}}}$$ of G. As $$G_{{\mathbb {C}}}/G^c$$ is non-compactly causal, it also contains three types of wedge domains. Our results says that the intersection of these domains with G/H agree with the wedge domains in G/H.

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In a paper entitled cocycles and spectra Helson and Par ry prove that for every T there exists a q~ such that V ~ has Lebesgue spectrum, moreover q~ can be chosen to be real, i.e., taking values + 1 and 1. The purpose of this paper is to extend this result to certain actions of countable groups which includes ergodic non-singular actions of countable abelian groups. We blend the method of Helson and Parry with the notions of weak equivalence and weak von Neumann transformations. In section 3 and 4 we discuss these results in connection with systems of imprimitivity. The problem of extending the result quoted above to countable groups was raised by H. Helson to one of us. It is a pleasure to acknowledge his interest and encouragement in this work.

Abért and Weiss have shown that the Bernoulli shift ${s}_{\Gamma } $ of a countably infinite group $\Gamma $ is weakly contained in any free measure preserving action $\boldsymbol{a}$ of $\Gamma $. Proving a conjecture of Ioana, we establish a strong version of this result by showing that ${\boldsymbol{s}}_{\Gamma } \times \boldsymbol{a}$ is weakly equivalent to $\boldsymbol{a}$. Using random Bernoulli shifts introduced by Abért, Glasner, and Virag, we generalize this to non-free actions, replacing ${\boldsymbol{s}}_{\Gamma } $ with a random Bernoulli shift associated to an invariant random subgroup, and replacing the product action with a relatively independent joining. The result for free actions is used along with the theory of Borel reducibility and Hjorth’s theory of turbulence to show that, on the weak equivalence class of a free measure preserving action, the equivalence relations of isomorphism, weak isomorphism, and unitary equivalence are not classifiable by countable structures. This in particular shows that there are no free weakly rigid actions, that is, actions whose weak equivalence class and isomorphism class coincide, answering negatively a question of Abért and Elek. We also answer a question of Kechris regarding two ergodic theoretic properties of residually finite groups. A countably infinite residually finite group $\Gamma $ is said to have property ${\text{EMD} }^{\ast } $ if the action ${\boldsymbol{p}}_{\Gamma } $ of $\Gamma $ on its profinite completion weakly contains all ergodic measure preserving actions of $\Gamma $, and $\Gamma $ is said to have property $\text{MD} $ if $\boldsymbol{\iota} \times {\boldsymbol{p}}_{\Gamma } $ weakly contains all measure preserving actions of $\Gamma $, where $\boldsymbol{\iota} $ denotes the identity action on a standard non-atomic probability space. Kechris has shown that ${\text{EMD} }^{\ast } $ implies $\text{MD} $ and asked if the two properties are actually equivalent. We provide a positive answer to this question by studying the relationship between convexity and weak containment in the space of measure preserving actions.

The first part of this paper deals with problems concerning the symmetric algebra of complex-valued polynomial functions on the complex vector space of n by k matrices. In this context, a generalization of the so-called “classical separation of variables theorem” for the symmetric algebra is obtained. The second part is devoted to the study of certain linear representations, on the above linear space (the symmetric algebra) and its subspaces, of the complex general linear group of order k and of its subgroups, namely, the unitary group, and the real and complex special orthogonal groups. The results of the first part lead to generalizations of several well-known theorems in the theory of group representations. The above representation, of the real special orthogonal group, which arises from the right action of this group on the underlying vector space (of the symmetric algebra) of matrices, possesses interesting properties when restricted to the Stiefel manifold. The latter is defined as the orbit (under the action of the real special orthogonal group) of the n by k matrix formed by the first n row vectors of the canonical basis of the k-dimensional real Euclidean space. Thus the last part of this paper is involved with questions in harmonic analysis on this Stiefel manifold. In particular, an interesting orthogonal decomposition of the complex Hilbert space consisting of all square-integrable functions on the Stiefel manifold is also obtained.

Congratulations to Professor Helland on the elegant development of optimal partial least squares (PLS) regression, which reconceptualizes model reduction through the geometry of group actions and Hilbert spaces. Within this framework, PLS identifies a lower-dimensional subspace of functions that is provably optimal among all m $$ m $$ -dimensional reductions under specified covariance or spectral conditions. This rigorous notion of optimality naturally raises the question: can an analogous concept be established for domain adaptation (Ganin et al., 2016; He et al., 2024; Luo et al., 2025)., a technique that has recently attracted significant attention in modern machine learning? To make the context of domain adaptation concrete, consider a simple example. Suppose we want to classify images of cats and dogs. In the source domain S $$ S $$ , we observe clean images; in the target domain T $$ T $$ , the same cats and dogs appear, but all images are rotated by arbitrary angles. A classifier trained on S $$ S $$ may fail badly in T $$ T $$ , because the marginal distributions of pixels, say, f S $$ {f}_S $$ and f T $$ {f}_T $$ , differ drastically. Yet the domains share invariant predictive features, such as those extracted by convolutional pooling layers that ignore rotations. If we can find a representation map h : 𝒳 → 𝒵 that captures these invariants, then prediction becomes possible across domains. This rotation example highlights the central challenge of domain adaptation: although f S $$ {f}_S $$ and f T $$ {f}_T $$ differ, a representation h $$ h $$ may capture outcome-relevant invariants across domains. The key question is whether one can rigorously define an optimal representation, balancing predictive accuracy with invariance, analogous to how PLS identifies optimal subspaces in regression. Moreover, what does it mean for a representation h $$ h $$ to be “optimal” for domain adaptation. This matters because, without a formal notion of optimality, existing methods such as adversarial training can only propose candidate solutions without offering a principled benchmark. The challenge arises from the fact that the distributions of x $$ \mathbf{x} $$ may differ arbitrarily across domains, making direct transfer of models infeasible and requiring careful extraction of predictive features that remain invariant. Helland's symmetry-based framework provides precisely the right tools for this task: group actions, Hilbert spaces, and invariance measures that together define optimality in rigorous geometric terms. In practice, adversarial training (Ganin et al., 2016) enforces h # f S ≈ h # f T $$ {h}_{\#}{f}_S\approx {h}_{\#}{f}_T $$ by jointly training a domain discriminator to distinguish between h # f S $$ {h}_{\#}{f}_S $$ and h # f T $$ {h}_{\#}{f}_T $$ and a representation map h $$ h $$ to both confuse the discriminator and minimize prediction loss on the source domain; however, optimality was not addressed. Our formulation (1), grounded in Helland's geometry, provides the missing rigor by defining “optimal” invariant representations as solutions to constrained optimization problems, in the same way that PLS characterizes optimal subspaces. In sum, the challenge in domain adaptation is that source and target domains differ in their raw covariates x $$ \mathbf{x} $$ , making direct transfer infeasible and necessitating extraction of invariant predictive features z $$ \mathbf{z} $$ . By embedding this problem into Helland's symmetry-based framework, we obtain a principled notion of optimality, that is, minimizing prediction risk subject to distributional invariance. This new formulation replaces heuristic criteria with a precise benchmark for optimality, clarifying why some representations are preferable and enabling fair comparison across methods. This also demonstrates the novel applicability of Helland's framework to modern machine learning. The article was supported by National Institutes of Health under grants R01CA249096 and R01CA269398. The authors declare no conflicts of interest. Data sharing not applicable to this article as no datasets were generated or analyzed during the current study.

We study an algebraic varieties with the action of a reductive group G. The relation is elucidated between the notions of complexity and rank of an arbitrary G-variety and the structure of stabilizers of general position of some actions of G itself and its Borel subgroup. The application of this theory to homogeneous spaces provides the explicit formulas for the rank and the complexity of quasiaffine G/H in terms of co-isotropy representation of H. The existence of Cartan subspace (and hence the freeness of algebra of invariants) for co-isotropy representation of a connected observable spherical subgroup H is proved.

We introduce a notion of a filtered model structure and use this notion to produce various model structures on procategories. We give several examples, including a homotopy theory for G-spaces, where G is a profinite group. The class of weak equivalences are an approximation to the class of underlying weak equivalences.

Introduction. 1. Hilbert's fifth problem and proper actions of Lie groups S. Illman. 2. Equivariant algebraic vector bundles over representations - a survey M. Masuda. 3. G-manifolds and G-vector bundles in algebraic, semialgebraic, and definable categories T. Kawakami. 4. Geometry of Finite Topological and equivariant finite topological spaces S. Kono, F. Ushitaki. 5. On the theory of homotopy representations. A survey I. Nagasaki. 6. Manifolds as fixed point sets of smooth compact Lie group actions K. Pawalowski. 7. Surgery and homotopy theory in study of the transformation groups Y. Kitada. 8. Kervaire's obstructions of free actions of finite cyclic groups on homotopy spheres Y. Kitada. 9. The Burnside ring revisited M. Morimoto. 10. Multiplicative stabilization and transformation groups S. Kwasik, R. Schultz. 11. Symmetries on manifolds, deformations and rational homotopy: a survey M. Raussen. 12. Rigidity of codimension one locally free actions of solvable Lie groups N. Tsuchiya, A. Yamakawa. 13. Smooth actions of non-compact semi-simple Lie groups F. Uchida, K. Mukoyama. 14. Hamiltonian group actions and equivariant indices T. Takakura. 15. Controlled methods in equivariant topology, a survey E. Pedersen. Index.

We study in this paper combinatorial problems concerning graphs generated by measure preserving actions of countable groups on standard measure spaces. In particular we study chromatic and independence numbers, in both the measure-theoretic and the Borel context, and relate the behavior of these parameters to properties of the acting group such as amenability, Kazhdan’s property (T), and freeness. We also prove a Borel analog of the classical Brooks’ Theorem in finite combinatorics for actions of groups with finitely many ends.

We establish criteria for turbulence in certain spaces of C*-algebra representations and apply this to the problem of nonclassifiability by countable structures for group actions on a standard atomless probability space (X,\mu) and on the hyperfinite II_1 factor R. We also prove that the conjugacy action on the space of free actions of a countably infinite amenable group on R is turbulent, and that the conjugacy action on the space of ergodic measure-preserving flows on (X,\mu) is generically turbulent.