Constructing groups with flat-rank greater than 1

Abstract: The contraction subgroup for $x$ in the locally compact group, $G$, $\mathop{con}(x) = \left\{ g\in G \mid x^ngx^{-n} \to 1\text{ as }n\to\infty \right\}$, and the Levi subgroup is $\mathop{lev}(x) = \left\{ g\in G \mid \{x^ngx^{-n}\}_{n\in\mathbb{Z}} \text{ has compact closure}\right\}$. The following will be shown.

Let $G$ be a totally disconnected, locally compact group and $x\in G$. Let $y\in{\sf lev}(x)$. Then there are $x'\in G$ and a compact subgroup, $K\leq G$ such that: $K$ is normalised by $x'$ and $y$, $\mathop{con}(x') = \mathop{con}(x)$ and $\mathop{lev}(x') = \mathop{lev}(x)$, and the group $\langle x',y,K\rangle$ is abelian modulo $K$, and hence flat.

If no compact open subgroup of $G$ normalised by $x$ and no compact open subgroup of $\mathop{lev}(x)$ normalised by $y$, then the flat-rank of $\langle x',y,K\rangle$ is equal to $2$.

group theory

Audience: researchers in the discipline

Symmetry in Newcastle