Flat groups of automorphisms of totally disconnected, locally compact groups

A group, $\fl{H}$, of automorphisms of a totally disconnected locally compact group, $G$, is flat if there is a compact open $U\leq G$ such that the index $[\alpha(U):U\cap \alpha(U)]$ is mininimized for every $\alpha\in\fl{H}$. The stabilizer of $U$ in $\fl{H}$ is a normal subgroup, $\fl{H}_u$; the quotient $\fl{H}/\fl{H}_u$ is a free abelian group; and the rank of $\fl{H}$ is the rank of this free abelian group. Each singly generated group $\langle\alpha\rangle$ is flat and has rank either $0$ or $1$. Higher rank groups may be seen in Lie groups over local fields and automorphism groups of buildings. Flat groups of automorphisms exhibit many of the features of these special examples, including analogues of roots and a factoring of $U$ into analogues of root subgroups. New proofs of improved versions of these results are presented here.