Lie Groups, Representations and Discrete Mathematics

Let $\alpha$ be an automorphism of a totally disconnected locally compact group $G$. There is a canonical form for $\alpha$ that partially fills the role played by the Jordan canonical form of $\mathrm{ad}( \alpha )$ in the case when $G$ is a Lie group. This general canonical form identifies structures in $G$ analogous to eigenspaces of $\alpha$ and associated positive integers analogous to absolute values of eigenvalues. Further work promises to draw closer analogies with the theory of Lie groups.

The canonical form carries information only when $G$ is non-discrete.  Nevertheless, it can be used to investigate finitely generated discrete groups via their cocompact embeddings into totally disconnected groups.