# Additive Conjugacy and the Bohr Compactification of Orthogonal Representations

## TL;DR

This paper introduces a new invariant for orthogonal representations of finitely generated groups based on additive conjugacy and the Bohr compactification, linking it to amenability and invariant vectors.

## Contribution

It defines additive conjugacy for representations and constructs a topological invariant via the Bohr compactification, revealing new connections to group amenability.

## Key findings

- Almost invariant vectors are additive conjugacy invariants.
- Amenability characterized by invariant homomorphisms from $L^2(G)$.
- The Bohr compactification encodes additive conjugacy information.

## Abstract

We say that two unitary or orthogonal representations of a finitely generated group $G$ are additive conjugates if they are intertwined by an additive map, which need not be continuous. We associate to each representation of $G$ a topological action that is a complete additive conjugacy invariant: the action of $G$ by group automorphisms on the Bohr compactification of the underlying Hilbert space. Using this construction we show that the property of having almost invariant vectors is an additive conjugacy invariant. As an application we show that $G$ is amenable if and only if there is a nonzero homomorphism from $L^2(G)$ into $\mathbb{R}/\mathbb{Z}$ that is invariant to the $G$-action.