Lattices in Higher Rank Semisimple Groups

• Lattices in higher rank semisimple groups are discrete subgroups with finite invariant measure that embody key rigidity and classification properties.
• They utilize techniques such as Margulis superrigidity, higher property T, and measure rigidity to establish cohomological vanishing and stability features.
• Insights include character rigidity, profinite volume invariance, and restrictions on orderability, linking arithmetic properties with dynamical systems.

A lattice in a higher rank semisimple group is a discrete subgroup whose coset space has finite invariant measure, and whose ambient group possesses a semisimple structure with real (or algebraic) rank at least two. The concept is central in the intersection of Lie theory, algebraic groups, ergodic theory, and rigidity phenomena, and serves as the paradigmatic subject for several rigidity and classification programs in modern mathematics.

1. Definitions and Fundamental Structures

Let $G$ be a connected semisimple Lie group (or a semisimple algebraic group over a local or global field) with finite center and without compact factors. The real rank of $G$, denoted $\mathrm{rank}_{\mathbb{R}}(G)$, is the dimension of a maximal real split torus. When $\mathrm{rank}_{\mathbb{R}}(G) \geq 2$, $G$ is designated as "higher rank."

A lattice $\Gamma < G$ is a discrete subgroup such that $G/\Gamma$ has finite $G$-invariant measure. Lattices are called irreducible if their projection to every proper simple factor of $G$ is dense.

In the non-Archimedean case, an analytic $k$-group $G$ (with $k$ a non-Archimedean local field) acts on its Bruhat–Tits building, with lattices defined analogously by the existence of a finite-volume quotient $G/\Gamma$ (Gelander et al., 2017).

2. Rigidity, Character Rigidity, and Higher Property T

(a) Superrigidity and Character Theory

Margulis superrigidity is foundational: every finite-dimensional complex representation of an irreducible higher-rank lattice virtually extends to a rational representation of the ambient group. This leads to pronounced character rigidity—for example, every extremal character (normalized class function, positive-definite and conjugation invariant) is either almost-periodic (from a finite-dimensional representation) or is the regular character (Bader et al., 2021, Dogon et al., 29 Jul 2025, Boutonnet et al., 2019). In non-uniform higher-rank lattices, any character is either regular or factors through a finite quotient (Dogon et al., 29 Jul 2025).

A structural noncommutative Nevo–Zimmer theorem—now established in arbitrary characteristic—implies that any ergodic von Neumann algebra over such a lattice either admits no nontrivial map to a boundary, or splits off a finite-dimensional algebraic quotient. The only nontrivial intermediate subalgebras in boundary crossed products correspond to boundary actions associated to parabolic subgroups (Bader et al., 2021).

(b) Higher Property T and Cohomology

For any irreducible lattice $\Gamma$ in a semisimple $F$-simple Lie group of rank $r \geq 2$, one has higher property $(T_{r-1})$; that is, $H^j(\Gamma, V) = 0$ for all unitary $\Gamma$-modules $V$ with $V^\Gamma=0$ and degrees $j < r$ (Bader et al., 25 Nov 2025). This cohomological rigidity has profound consequences: operator algebraic vanishing, fixed-point properties for low-dimensional CAT(0) actions (Serre–Farb property $FA_{r-1}$), and far-reaching geometric expansion properties.

Formally, for a discrete group $\Gamma$ of type $FP_\infty$, $\Gamma$ has property $(T_n)$ iff certain cohomology groups with coefficients in its group C*-algebra, or its Kazhdan-deflated component, vanish in degrees up to $n$ and are Hausdorff in degree $n+1$ (Bader et al., 25 Nov 2025). For lattices, these statements generalize property (T), with significant applications to $L^p$-cohomology, Banach space representations, and the structure of invariant random subgroups.

3. Measure Rigidity and Dynamical Classifications

Structural measure rigidity for smooth actions of higher-rank lattices on manifolds is governed by dimension invariants $r(G)$ (critical) and $m(G)$ (intermediate), computed from the restricted–root data of the Lie algebra (Brown et al., 2016):

• If $\dim M < r(G)$, any $C^{1+\beta}$ action of $\Gamma$ on compact $M$ preserves a probability measure.
• If $\dim M = r(G)$, any such action either preserves measure, or is measurably isomorphic to a standard boundary action over a homogeneous space $Q\backslash G$ for maximal parabolic $Q$.
• If $\dim M \leq m(G)$, the action is measurably isomorphic to a relatively measure-preserving skew-product over such a boundary action.

Proof employs suspension bundles, amenable-boundary measure construction, Lyapunov exponent analysis, and (in the critical/intermediate dimension case) entropy-theoretic invariance forcing projective factors. These theorems underpin the Zimmer program, providing sharp rigidity for low-dimensional lattice actions.

A smooth rigidity theorem further demonstrates that any $C^\infty$ factor of the standard projective action (on a flag manifold) admitting a differentiable sink is $C^\infty$-conjugate to the standard action on a flag manifold $G/Q$ (Gorodnik et al., 2016).

4. Representations, Stability, and Orderability

(a) Unitary and Rank-Metric Stability

Contrast emerges between stability in various senses:

• Uniform stability: Higher-rank lattices admit uniform stability for finite-dimensional unitary almost-representations (in submultiplicative norms), as established by showing vanishing of an “asymptotic” second cohomology for such modules (Glebsky et al., 2023).
• Rank-metric instability: In contrast, higher-rank lattices fail strict uniform stability in the normalized rank metric: there exist arbitrarily good almost-representations which cannot be approximated by honest representations at small normalized rank distance (Bachner, 2024). This arises from compression of highest-weight representations, the Borel density theorem, and superrigidity, and marks a stark distinction from amenable groups.

(b) Orderability and Actions on One-Manifolds

Lattices in higher-rank semisimple groups are not left-orderable. No irreducible lattice in such $G$ admits a left-invariant total order, or equivalently, any action by orientation-preserving homeomorphisms on $\mathbb{R}$ is trivial (Morris, 2024, Deroin et al., 2020). The denial of left-orderability reflects extreme one-dimensional rigidity, in sharp contrast to the situation for surface groups or rank-1 lattices.

(c) Actions on Free and RAAG Automorphism Groups

Any homomorphism from a higher-rank irreducible lattice to $\mathrm{Out}(F_n)$, the outer automorphism group of a finitely generated free group, must have finite image (Bridson et al., 2010). For RAAG (right-angled Artin groups) automorphism groups, a similar phenomenon holds up to explicit rank bounds: infinite image homomorphisms can only exist if the real rank of the lattice is at most the maximal "SL-dimension" of the RAAG (Wade, 2011).

5. Invariant Random Subgroups, Local Rigidity, and Approximate Lattices

The theory of invariant random subgroups (IRS) for higher-rank lattices demonstrates that limits of sequence of such lattices, in the Chabauty topology, accumulate only on central subgroups: IRS rigidity and local structural rigidity extend to both real and non-Archimedean analytic groups (Gelander et al., 2017). Classifications show that for sequence of pairwise non-conjugate irreducible lattices, the normalized Betti numbers and relative Plancherel measures converge to their ambient group analogues.

For non-uniform higher-rank lattices, stabilizer rigidity for ergodic probability-preserving actions holds: either the action is essentially transitive, or stabilizers are almost surely trivial, confirming the Stuck–Zimmer conjecture (Dogon et al., 29 Jul 2025, Creutz, 2013).

In the setting of approximate subgroups, any strong approximate lattice in a higher-rank semisimple group must arise via arithmetic constructions, extending the rigidity paradigm beyond genuine subgroups (Machado, 2020).

6. Arithmeticity, Discreteness, and Profinite Invariants

All higher-rank irreducible lattices are arithmetic, enabling powerful connections to number theory. For cocompact lattices, Margulis’ uniform discreteness conjecture is shown equivalent to Lehmer’s conjecture on Mahler measures of algebraic integers (Pham et al., 2020). The Mahler measure provides a bridge between geometric discreteness and algebraic properties of matrix eigenvalues.

A fundamental recent result establishes that, assuming the congruence subgroup property (CSP), the profinite completion of a lattice in higher-rank semisimple Lie groups determines its covolume with respect to the renormalized Killing measure (Kammeyer et al., 2024). This profinite rigidity signifies that the arithmetic and measure-theoretic complexity of $\Gamma$ is fully encoded in its finite quotients—a phenomenon not shared by rank-one arithmetic lattices.

Property	Higher-Rank Lattices ($\text{rank} \geq 2$)	Rank-One Lattices
Superrigidity	Holds (Margulis)	Fails in general
Character rigidity	Complete: regular or finite image	Infinite-dimensional characters possible
Uniform stability (unitary)	Yes (Glebsky et al., 2023)	No (rank-one fails)
Rank-metric stability	No (strict) (Bachner, 2024)	Open/mixed
Left-orderability	Not left-orderable	Often left-orderable
Profinite volume invariance	Yes with CSP (Kammeyer et al., 2024)	Open (e.g., for hyperbolic 3-manifolds)

7. Research Directions and Open Problems

Current open directions include the flexible rank-metric stability problem for higher-rank lattices—whether allowing slight increases in target dimension recovers stability lost in the strict sense (Bachner, 2024). For operator-algebraic, dynamical, and geometric rigidity, further extensions to lattices without property (T), as well as deeper understanding of low-rank and positive-characteristic settings, remain at the forefront.

There is ongoing investigation into the classification and structure of approximate lattices in rank-one and amenable groups (Machado, 2020). The relationship between homological invariants, cohomological vanishing, and geometric expansion (e.g., cosystolic expansion) continues to bridge rigidity theory with topological and geometric group theory (Bader et al., 25 Nov 2025).

Research increasingly leverages operator-algebraic frameworks—such as von Neumann algebras, boundary theory, and noncommutative ergodic theorems—to provide unified classification schemas for characters, invariant random subgroups, and dynamical actions (Bader et al., 2021, Dogon et al., 29 Jul 2025). The operator-algebraic Margulis factor theorem gives a definitive list of intermediate von Neumann subalgebras in boundary crossed product factors, indexed by parabolic subgroups.

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References:

For key foundational and recent advances, see (Brown et al., 2016, Bader et al., 25 Nov 2025, Bader et al., 2021, Creutz, 2013, Kammeyer et al., 2024, Bachner, 2024, Dogon et al., 29 Jul 2025, Gelander et al., 2017, Glebsky et al., 2023, Bridson et al., 2010, Wade, 2011, Gorodnik et al., 2016, Machado, 2020, Boutonnet et al., 2019, Pham et al., 2020, Morris, 2024, Deroin et al., 2020).