Flatmates and the bounded cohomology of algebraic groups

Technical Summary: Flatmates and the Bounded Cohomology of Algebraic Groups

Problem Statement
The primary objective of this work is to establish the vanishing of continuous bounded cohomology for algebraic groups over non-Archimedean local fields. Specifically, the paper addresses the following gap: while the ordinary continuous cohomology of simple algebraic groups over such fields is known to vanish in positive degrees (a classical result by Garland, Casselman, Wigner, and Harder), the behavior of bounded cohomology remained unknown.

This problem is situated within a broader context of understanding bounded cohomology for Lie groups. While Dupont's conjecture (that real continuous cohomology classes of connected simple Lie groups are bounded) remains open for many groups, the non-Archimedean case presents a distinct challenge. In the ordinary setting, vanishing above the rank is trivial because the associated Bruhat–Tits building is a contractible complex of that dimension. However, this "soft" principle fails for bounded cohomology; for instance, the tree associated with $SL_2(\mathbb{Q}_p)$ is contractible, yet it admits non-trivial degree-two bounded cohomology. The paper aims to prove that, contrary to the behavior of "almost-cocycles" in other contexts, the bounded cohomology of these groups vanishes entirely in all positive degrees.

Methodology
The proof strategy relies on a reduction from algebraic groups to the geometry of their associated Bruhat–Tits buildings, followed by a novel combinatorial approach to the bounded cohomology of these geometric structures.

1. Reduction to Buildings: The vanishing theorem for algebraic groups (Theorem A) is reduced to a statement about the automorphism groups of Bruhat–Tits buildings (Theorem B). Using structure theory, the problem is reduced to the case of simple groups acting strongly transitively on locally finite Euclidean buildings.
2. The Flatmate Complex: The core technical innovation is the analysis of the "flatmate complex" (also termed the cokotcomplex). This simplicial complex consists of all tuples of vertices that lie within a common apartment (flat) of the building. The authors previously proposed this object in the context of trees [BM19] but faced significant combinatorial difficulties in generalizing it to higher-rank buildings.
3. Simplicial Tools and Support Control: To overcome the combinatorial complexity of arbitrary point configurations in buildings, the paper introduces a set of simplicial tools:
   • Support Control: A principle leveraging the $\ell^1$-norm on homology cycles. It establishes that if every finite subcomplex with $n$ vertices is contained in an acyclic subcomplex with at most $\phi(n)$ vertices, the complex is uniformly acyclic (admitting a bounded contracting homotopy).
   • Uniform Nerve Principles: The authors develop a uniform version of the nerve lemma. They prove that if a cover of a simplicial complex consists of hypersimplices (and their intersections are hypersimplices), the complex is uniformly homotopy equivalent to the nerve of the cover, with bounds depending only on the homology degree.
4. Application to Buildings: The authors apply these tools to the apartment system of a Euclidean building. They demonstrate that the nerve of the apartment system is uniformly acyclic. By the uniform nerve principle, this implies the flatmate complex itself is uniformly acyclic.

Key Results

• Theorem A (Algebraic Groups): For any algebraic group $G$ over a non-Archimedean local field $k$, the continuous bounded cohomology of $G(k)$ with real coefficients vanishes in every positive degree.
• Theorem B (Buildings): If a locally compact group $G$ acts properly and strongly transitively by automorphisms on a locally finite Euclidean building, its continuous bounded cohomology vanishes in every positive degree.
• Theorem F (Flatmates): The flatmate complex of any discrete irreducible Euclidean building is uniformly acyclic. This resolves the "flatmate conjecture" raised in earlier work [BM19].
• Theorem D & Dbis (Arithmetic Groups): The paper derives vanishing and invariance theorems for $S$-arithmetic groups.
  • If $G$ is anisotropic over Archimedean completions, the bounded cohomology of the $S$-arithmetic group $\Gamma$ vanishes in degrees $0 < n < 2 \sum_{v \in S} \text{rank}_{K_v}(G)$.
  • If isotropic Archimedean places are present, the bounded cohomology of $\Gamma$ is isomorphic to that of the product of the corresponding Archimedean Lie groups in the same range of degrees.

Significance and Claims
The paper claims that Theorem A provides a strengthening of classical vanishing theorems for non-Archimedean groups. While ordinary cohomology vanishing is well-understood, the vanishing of bounded cohomology is a "strict strengthening," implying that even "almost-cocycles" (such as quasi-morphisms in degree 2) must be trivial.

The authors position their work as a solution to the non-Archimedean analogue of the conjecture regarding the isomorphism between bounded and ordinary cohomology for semi-simple Lie groups. Since ordinary cohomology vanishes for these groups, the result implies the bounded cohomology also vanishes, a fact previously known only for rank-one groups (trees).

Furthermore, the paper introduces the "support control" principle and the "uniform nerve principle" as general simplicial tools. The authors suggest these methods are of independent interest and may be applicable beyond algebraic groups, particularly in contexts involving the glueing of complexes or the study of large transformation groups.

The results also complete a missing piece in the study of bounded cohomology for $S$-arithmetic groups, allowing for the derivation of invariance theorems that relate the cohomology of discrete arithmetic groups to the continuous cohomology of their ambient Lie groups, extending previous results that were limited to degree two or rank-one cases.