Higher property T and below-rank phenomena of lattices

Imagine you are trying to understand the "personality" of a massive, complex machine (a mathematical group). Some machines are rigid; if you try to wiggle them, they resist. Others are flexible and wobble easily.

This paper, written by Uri Bader and Roman Sauer, is about discovering a new, super-powerful kind of rigidity called "Higher Property T."

Here is the breakdown using simple analogies:

1. The Basics: What is "Property T"?

Think of a group as a team of dancers.

Normal Property T (Kazhdan's Property T): Imagine the dancers are holding hands in a very tight circle. If someone tries to push the circle from the outside, the whole group moves together as one solid block. They don't wobble. In math terms, this means the group resists "almost" moving; it forces them to either move perfectly or not at all. This is a famous property discovered by mathematician David Kazhdan.
The "Rank": Think of the "Rank" of a group as the number of independent directions the machine can move in. A simple machine might only move forward/backward (Rank 1). A complex machine might move forward, sideways, up, down, and rotate (High Rank).

2. The Big Discovery: "Higher Property T"

The authors ask: What if a group is rigid not just in one direction, but in many complex ways at once?

They define Higher Property T (specifically $(T_{r-1})$ ) as a group that is rigid in all directions up to its Rank minus one.

The Analogy: Imagine a spiderweb.
- A normal "Property T" web is so tight that if you poke it, the whole thing vibrates instantly.
- A "Higher Property T" web is so incredibly stiff that if you poke it, twist it, or stretch it in any way up to a certain complexity, it refuses to deform. It acts like a solid piece of steel rather than a web.

The Main Result (Theorem 1):
The authors prove that if you take a "lattice" (a regular, repeating pattern of points) inside a high-rank machine (a semisimple Lie group), that pattern has this super-rigidity.

If the machine has Rank $r$ , the pattern is rigid in $r-1$ different complex ways.
This is like saying a crystal lattice inside a high-dimensional space is unbreakable in almost every direction you can think of.

3. The "Below-Rank" Phenomena

The paper explores what happens when you look at these groups in dimensions lower than their maximum capacity (below the rank).

The "Silence" of the Group: In mathematics, "cohomology" is a way of measuring holes or gaps in a shape. Usually, complex shapes have many holes.
The Discovery: The authors show that for these high-rank lattices, the "holes" vanish completely in dimensions below the rank.
The Metaphor: Imagine a sponge. Usually, a sponge has holes everywhere. But these special high-rank lattices are like a sponge that has magically filled in all its holes up to a certain size. They become "solid" and "hole-free" in those lower dimensions.

4. Why Does This Matter? (The Real-World Connection)

You might ask, "Who cares about rigid mathematical groups?" The paper connects this to several deep mysteries:

Stability: If a system is "rigid," it is stable. This helps mathematicians understand why certain structures (like the symmetries of space) don't fall apart when slightly disturbed.
Expansion: Think of a social network. If everyone is connected in a specific "rigid" way, information spreads instantly and efficiently. These groups act like perfect "expanders," which is useful for computer science and cryptography.
Geometry and Shapes: The rigidity forces these groups to behave in very predictable ways when they act on shapes (like spheres or hyperbolic planes). This helps solve problems about how shapes can be folded or stretched without tearing.

5. The "Conjectural Framework" (The Unfinished Puzzle)

The paper doesn't just present facts; it draws a map of what we think is true but haven't proven yet.

They propose that this rigidity works even with "weird" types of numbers (Banach spaces), not just the standard ones.
They connect this to a "Spectral Gap" (a gap in the energy levels of the system). If the gap is wide enough, the group is rigid.
They suggest that if you have a group with this super-rigidity, it cannot be broken down into smaller, simpler pieces easily. It's an "atomic" unit of mathematical structure.

Summary in One Sentence

This paper proves that high-dimensional mathematical lattices are incredibly stiff and "hole-free" in many directions, and it proposes a grand theory that this stiffness explains why these groups are so stable, rigid, and resistant to change, connecting abstract algebra to geometry, topology, and even computer science.

The "Takeaway" Metaphor:
If standard math groups are like a wobbly Jell-O mold, this paper shows that high-rank lattices are like a diamond. Not only is the diamond hard, but the authors discovered that it is impossible to scratch it, dent it, or break it in any direction up to a certain depth. They then spent the paper mapping out exactly how this "diamond hardness" affects everything else in the mathematical universe.

Here is a detailed technical summary of the paper "Higher Property T and Below-Rank Phenomena of Lattices" by Uri Bader and Roman Sauer.

1. Problem Statement and Motivation

The paper addresses the generalization of Kazhdan's Property T to higher degrees, termed Higher Property T (denoted as $(T_n)$ and $[T_n]$ ), and investigates its relationship with "below-rank phenomena" in semisimple Lie groups and their lattices.

Context: Classical Property T ( $T_1$ ) is a rigidity property where a group has no almost invariant vectors in unitary representations without invariant vectors. It is known that lattices in simple Lie groups of rank $r$ satisfy Property T.
The Gap: While higher-rank lattices are known to satisfy Property T, the behavior of their cohomology in degrees $1 < j < r$ (below the rank) was less understood, particularly regarding general coefficients (Banach spaces, von Neumann algebras) and non-uniform lattices.
Core Question: Do lattices in simple groups of rank $r$ satisfy a higher analogue of Property T, specifically the vanishing of cohomology $H^j(\Gamma, V) = 0$ for $j < r$ for various classes of coefficients $V$ ? Furthermore, how does this relate to geometric and rigidity phenomena occurring below the rank?

2. Methodology

The authors employ a dual approach, combining abstract operator-algebraic techniques with geometric group theory and representation theory.

A. Abstract Group-Theoretic Framework (Section 2)

The authors treat Higher Property T as an intrinsic property of abstract groups, independent of their realization as lattices.

Operator Algebraic Characterizations: They reformulate $(T_n)$ $(T_{n})$ and $[T_n]$ $[T_{n}]$ using the cohomology of the group with coefficients in the maximal $C^*$ -algebra ( $C^*\Gamma$ $C^{*} Γ$ ) and the von Neumann algebra ( $W^*\Gamma$ $W^{*} Γ$ ).
- They establish that $\Gamma$ has $[T_n]$ if and only if $H^k(\Gamma, C^*\Gamma) = 0$ for $1 \le k \le n $and$ H^{n+1}(\Gamma, C^*\Gamma)$ is Hausdorff.
- They provide a sum-of-squares characterization for $(T_n)$ involving the Laplace operators $\Delta_k$ on the group ring, generalizing Ozawa's criterion for classical Property T.
Ultrapowers and Banach Modules: The paper utilizes ultrapowers of Banach spaces to analyze the Hausdorffness of cohomology groups. They prove that if cohomology vanishes for all ultrapowers, it vanishes for the original space under specific conditions.
Finiteness Properties: The analysis assumes groups satisfy the finiteness property $FP_\infty(\mathbb{Q})$ to ensure well-behaved resolutions, though they discuss the necessity of these assumptions.

B. Lattices in Semisimple Groups (Section 3)

The authors apply the abstract framework to lattices $\Gamma < G$ , where $G$ is a semisimple Lie group of rank $r$ .

Induction on Rank: They use a cohomological induction argument based on the Opposition Complex of the Tits building. The stabilizers of simplices in this complex are Levi subgroups of lower rank, allowing an inductive proof of vanishing results.
Polynomial Cohomology: To handle non-uniform lattices (where the standard Shapiro Lemma fails), they utilize polynomial cohomology ( $H^*_{pol}$ ). They leverage results by Leuzinger and Young showing that filling functions for these lattices are polynomial below the rank, allowing a comparison between polynomial and standard cohomology.
Spectral Gap Conjectures: The proofs for general Banach coefficients rely on a "Strong Spectral Gap Conjecture" (Conjecture 68) regarding the action of rank-1 subgroups on super-reflexive Banach spaces.

3. Key Contributions and Results

A. Main Theorems on Higher Property T

Theorem 1 (Generalization of Kazhdan): Let $G$ be a simple algebraic group of rank $r$ over a local field $F$ , and $\Gamma < G$ a lattice.
- $\Gamma$ has property $(T_{r-1})$ .
- If $F$ is non-archimedean, $\Gamma$ has property $[T_{r-1}]$ .
- Note: This confirms that lattices in rank $r$ groups have vanishing cohomology up to degree $r-1$ for unitary representations without invariant vectors.
Theorem 11 & Corollary 12 (Banach and $L^p$ Coefficients):
- For a lattice $\Gamma$ in a simple Lie group of rank $r$ , and for $k \le r-1$ , the inclusion of invariants $L^p(M)^\Gamma \hookrightarrow L^p(M)$ induces isomorphisms in cohomology for any von Neumann algebra $M$ and $1 \le p < \infty$.
- Specifically, if $L^p(M)^\Gamma = 0$ , then $H^k(\Gamma, L^p(M)) = 0$ for all $k \le r-1$ .
- This extends the vanishing results to super-reflexive Banach spaces (conditional on Conjecture 68) and unconditionally for $L^p$ spaces ($1 < p < \infty $) and$ p=1$ (via specific fixed-point theorems).
Theorem 91 (Farb's Conjecture): Latties in simple groups of rank $r$ satisfy Property $FA_{r-1}$ (any cellular action on an $n$ -dimensional CAT(0) complex with $n < r$ has a global fixed point). This is derived from the $(T_n)_{L^1}$ property.

B. Conjectural Framework and Implications

The paper outlines a unified framework connecting Higher Property T to various rigidity phenomena:

Conjecture 7 (Super-reflexive Property T): $\Gamma$ has vanishing cohomology $H^j(\Gamma, V) = 0$ for $j < r$ for all super-reflexive Banach spaces $V$ with no invariants.
Conjecture 8 (Semisimple Case): For irreducible lattices in $S$ -semisimple groups, cohomology below the rank is isomorphic to the cohomology of the ambient group restricted to the invariant subspace.
Degree 1 Applications (Spectral Gap): The vanishing of $H^1$ $H^{1}$ is equivalent to Shalom's Spectral Gap Conjecture (Conjecture 105). Proving this would imply:
- Property $\tau$ (Lubotzky).
- Coamenability rigidity (Coamenable subgroups are finite index).
- IRS/URS Rigidity (Invariant Random Subgroups are supported on finite sets).
- Character Rigidity (Connes' conjecture: characters are central or finite-dimensional).
Degree 2 Applications (Stability): Cohomological vanishing in degree 2 relates to the stability of groups under perturbations and the non-approximability of certain central extensions (e.g., symplectic lattices are not Frobenius-approximable).

C. Geometric Phenomena

Waist Inequalities: The authors connect $(T_n)_{L^1}$ to uniform waist inequalities for families of finite covers of manifolds. If a group has $(T_n)_{L^1}$ , its finite covers exhibit topological overlap properties and waist inequalities in codimension $n$ .
Torsion Growth: They speculate on the vanishing of homology torsion growth for lattices below the rank, linking it to the "fixed price" conjecture.

4. Significance and Impact

Unification of Rigidity: The paper provides a cohesive theoretical framework linking algebraic cohomology, operator algebras, and geometric rigidity. It demonstrates that "Higher Property T" is the unifying mechanism behind various "below-rank" phenomena (vanishing cohomology, fixed point properties, spectral gaps).
Extension to Banach Spaces: By moving beyond Hilbert spaces to $L^p$ and super-reflexive Banach spaces, the authors significantly broaden the scope of rigidity theory. This is crucial for applications in geometric group theory where $L^p$ norms are natural.
Resolution of Open Problems:
- It proves Farb's Conjecture on Property $FA_{r-1}$ .
- It settles the cohomological vanishing for $L^1$ coefficients (Corollary 12), resolving a gap left by previous work on $p=1$ .
- It constructs uncountable families of simple Kazhdan groups with distinct $\ell^2$ -Betti numbers (Theorem 115), demonstrating a rich diversity of measurable structures even among rigid groups.
New Tools: The introduction of polynomial cohomology as a bridge for non-uniform lattices and the use of sum-of-squares criteria for higher property T provide new computational and theoretical tools for the field.
Future Directions: The paper clearly delineates the boundary between known results and open conjectures (e.g., the Strong Spectral Gap Conjecture), setting a roadmap for future research in the rigidity of lattices and the structure of infinite groups.

In summary, Bader and Sauer establish that Higher Property T is a robust, abstract group-theoretic property that governs the cohomological and geometric behavior of lattices in semisimple groups well below their rank, offering deep insights into the structure of these groups and their representations.