Flatmates and the bounded cohomology of algebraic groups

TL;DR

This paper proves that bounded cohomology vanishes for all algebraic groups over non-Archimedean local fields, using automorphism groups of Bruhat--Tits buildings and the flatmate conjecture, leading to new invariance results for arithmetic groups.

Contribution

It establishes the vanishing of bounded cohomology for algebraic groups over non-Archimedean fields, solving the flatmate conjecture and deriving related invariance theorems.

Findings

Bounded cohomology vanishes for all algebraic groups over non-Archimedean local fields.

The flatmate conjecture is solved, underpinning the main results.

Invariance theorems for arithmetic groups are established.

Abstract

For all algebraic groups over non-Archimedean local fields, the bounded cohomology vanishes. This follows from the corresponding statement for automorphism groups of Bruhat--Tits buildings, which hinges on the solution to the flatmate conjecture raised in earlier work with Bucher. Vanishing and invariance theorems for arithmetic groups are derived.