Non-Measure Equivalent Countable Groups

• Non-measure equivalent countable groups are discrete groups lacking a common σ-finite measure space coupling, defined by unique rigidity properties.
• Their classification relies on invariants like amenability, property (T), and ℓ²-Betti numbers, which differentiate rigid and flexible group structures.
• Advanced techniques such as cocycle and infinite product rigidity enable the construction of infinite and continuum families of pairwise non-ME groups.

Non-measure equivalent countable groups are pairs or families of countable discrete groups that are not measure equivalent (ME): there is no standard σ-finite measure space equipped with commuting, measure-preserving actions of both groups such that each action admits a finite-measure fundamental domain. The structure of ME-classes, as illuminated by recent advances, reveals deep rigidity and flexibility phenomena in group theory, ergodic theory, and measured groupoids. The existence of infinite and even continuum-sized families of pairwise non-ME groups—especially among groups with strong rigidity features such as property (T)—connects geometric, analytic, and representation-theoretic invariants in the classification of countable groups.

1. Measure Equivalence: Definition and Fundamental Properties

Two countable discrete groups $G$ and $H$ are measure equivalent if there exists a σ-finite measure space $(\Omega, m)$ and commuting, measure-preserving actions $G \times H \curvearrowright (\Omega, m)$ with each action admitting a finite-measure fundamental domain. This enables the decomposition of $\Omega$ into disjoint translates of subsets $X$ and $Y$ under $G$ and $H$ respectively, with $0 < m(X), m(Y) < \infty$, up to null sets. The ME-coupling preserves probabilistic and orbit structure information across groups, facilitating the transfer and comparison of ergodic-theoretic invariants. For discrete countable groups, ME is deeply linked to stable orbit equivalence of free ergodic pmp (probability measure preserving) actions (Ioana et al., 4 Dec 2025, Koivisto et al., 2017).

Certain group properties—most notably amenability and Kazhdan’s property (T)—are invariants under measure equivalence for unimodular, locally compact second countable (lcsc) groups, and hence for discrete groups. Non-isomorphic groups within the same ME-class share such invariants, strongly restricting possible ME equivalences across different rigidity types (Koivisto et al., 2017).

2. Rigidity Phenomena and ME-invariants

Key rigidity phenomena for ME derive from the invariance of properties such as amenability and property (T). For instance, all infinite countable amenable groups are measure equivalent to each other via Ornstein–Weiss theory, but no amenable group is ME to a nonamenable group (Koivisto et al., 2017). Property (T) is also invariant: nonamenable property (T) groups form their own ME-class, disjoint from both amenable and non-T groups. This stratifies the landscape of countable groups into ME-towers separated by fundamental rigidity properties.

Further ME-invariants include the sequences of $\ell^2$-Betti numbers and, in suitable rigidity contexts, group cohomology and cost. The $\ell^2$-Betti sequence $(\beta_n^{(2)}(G))_{n\ge0}$ is invariant up to scaling under ME, and vanishing/nonvanishing patterns in Betti numbers often form the technical backbone for distinguishing non-ME families (Neumann, 2021).

3. Explicit Infinite and Continuum Families of Non-ME Groups

Constructing explicit (uncountable) collections of pairwise non-ME countable groups is a central recent achievement. Two primary methods have emerged:

A. Continuum Families of Torsion-Free Property (T) Groups with Zero $\ell^2$-Betti Numbers

A construction (2025) produces a continuum-sized family $\{G_x\}_{x\in\{0,1\}^\mathbb{N}}$ where each $G_x$ is finitely generated, torsion-free, has property (T), and trivial $\ell^2$-Betti numbers in all degrees. The construction proceeds as follows (Ioana et al., 4 Dec 2025):

• For each $x \in \{0,1\}^\mathbb{N}$, define $N_x = \{k : x(k) = 1\}$.
• Set $D_x = \bigoplus_{k \in N_x} A_k$, where $\{A_k\}$ is a countable family of ICC, non-ME, $n$-generated, class $\mathcal{C}$ groups with vanishing $\ell^2$-Betti numbers.
• Build $G_x$ as a wreath-like extension involving a fixed torsion-free hyperbolic group $B$:

$1 \to D_x^{(B)} \to G_x \to B \to 1.$

Pairwise non-ME follows from infinite-product rigidity: $G_x \sim_\mathrm{ME} G_y \implies N_x = N_y$, so $x \neq y$ implies $G_x \not\sim_\mathrm{ME} G_y$.

B. Infinite Families of Simple Kazhdan Groups with Distinct $\ell^2$-Betti Profiles

Families of infinite, finitely presented, simple groups $\{\Gamma_n\}$ with property (T) are constructed such that for each $n$, $\beta_{2n}^{(2)}(\Gamma_n) > 0$ and all other higher Betti numbers vanish. As $\ell^2$-Betti sequences are ME-invariants (up to scaling), these groups are pairwise non-measure equivalent: distinct indices of nonvanishing preclude measure equivalence. The construction utilizes Kac–Moody lattices acting on products of buildings and Weyl-group combinatorics (Neumann, 2021).

Construction	Key Properties	Mechanism of Non-ME
$\{G_x\}_{x}$	Torsion-free, (T), zero $\ell^2$-Betti	Infinite-product rigidity and cocycle rigidity
$\{\Gamma_n\}$	Simple, (T), specific $\ell^2$-Betti pattern	Invariant $\ell^2$-Betti degrees

4. Classification in Special Classes: Right-Angled Artin Groups and Beyond

Within certain algebraic classes, ME-classifications have been made precise:

• For transvection-free right-angled Artin groups (RAAGs), two such groups are measure equivalent if and only if their extension graphs are isomorphic. For RAAGs with finite outer automorphism groups, measure equivalence further reduces to group isomorphism. Thus, in this class, ME-rigidity mirrors quasi-isometric rigidity (Horbez et al., 2020).
• Despite this rigidity, flexibility arises: any RAAG is measure equivalent to graph products of infinite amenable groups over the same graph; likewise, lattices in certain automorphism groups of universal covers introduce infinitely generated, non-uniform lattices ME to the original group. Therefore, RAAGs are not superrigid for ME in the strongest sense.

5. Techniques: Cocycle and Product Rigidity, Extensions, and Couplings

Two technical pillars underlie many modern non-ME constructions:

• Cocycle Rigidity: In various group extensions, ME-cocycles (arising in the decomposition of ME-couplings) are forced, by group structure and normal subgroup arrangements, to be “essentially trivial” on certain subgroups. Adjustments to fundamental domains then reduce ME-couplings to those of subgroups, leveraging known rigidity at this level (Ioana et al., 4 Dec 2025).
• Infinite Product Rigidity: When the Monod–Shalom product rigidity class $\mathcal{C}$ is available, countable direct sums of ICC-class groups exhibit rigidity: $\bigoplus_{k\in I}A_k$ is ME to $\bigoplus_{k\in J}A'_k$ only if there is a bijection $I \cong J$ matching ME-classes (Ioana et al., 4 Dec 2025).

Short exact sequences and wreath-like product constructions, especially over hyperbolic or property (T) groups, allow the transfer and amplification of subgroup-level rigidity to the full group, generating continuum-wide non-ME phenomena.

6. $\ell^2$-Betti Numbers and Quantitative Invariants

The vanishing or nonvanishing pattern of $\ell^2$-Betti numbers is a central ME-invariant. For countable groups $\Gamma$, the numbers $\beta_p^{(2)}(\Gamma)$, defined via the Murray–von Neumann dimension of the reduced $\ell^2$-homology, are preserved up to scaling in ME. This property makes them effective discriminants in the construction of non-ME families (Neumann, 2021, Ioana et al., 4 Dec 2025).

For example, the wreath-like groups $G_x$ satisfy $\beta_p^{(2)}(G_x)=0$ for all $p \geq 1$, while the Kac–Moody lattice groups $\Gamma_n$ are constructed so that $\beta_{2n}^{(2)}(\Gamma_n)>0$ but $\beta_k^{(2)}(\Gamma_n)=0$ for $k>2n$. Thus, for each of these families, the collection of $\ell^2$-Betti numbers is sufficient to distinguish ME-classes in the absence of proportional sequences.

7. Significance, Open Directions, and Further Invariants

The construction of continuum-sized, pairwise non-ME families among finitely generated, torsion-free, property (T) groups with prescribed $\ell^2$-Betti vanishing provides a decisive advance in the understanding of ME-class structure. It demonstrates that even in the most rigid and analytically constrained group-theoretic settings, the space of ME-classes can be as large as the continuum (Ioana et al., 4 Dec 2025).

Ongoing research directions include:

• Extending continuum non-ME constructions to further rigidity classes such as nonuniform lattices and mapping class groups.
• Quantitative analysis of ME-invariants (e.g., ergodic dimension, costs) in complex constructions such as wreath-like products.
• Investigating von Neumann equivalence (II$_1$-factor couplings) as a yet finer classification among such groups.

These advances underscore the richness of the ME landscape, with the interplay between analytic invariants, cocycle rigidity, and group extension theory as central organizing themes.