Diagram groups and groups of piecewise linear homeomorphisms of the line with global fixed points

Abstract

Assume $n \geq 2$ and $\ell = (r_{1}, \ldots, r_{k}) \in [0,1]^{k}$ is an increasing sequence of real numbers. Let $G_{n,\ell}$ denote the group of orientation-preserving piecewise linear homeomorphisms $h$ of $I = [r_{1}, r_{k}]$ such that: (i) $h'(x)$ is a power of $n$ where it is defined; (ii) if $h'(x)$ is undefined, then $x$ is an $n$-adic rational number, (iii) $h$ fixes each entry of $\ell$, and (iv) $h(\mathbb{Z}[1/n] \cap I) = \mathbb{Z}[1/n] \cap I$. We prove that $G_{n,\ell}$ is a diagram group $D(\mathcal{P}_{n,\ell}, \omega_{n,\ell})$ for all integers $n \geq 2$ and for all finite sequences $\ell$. The semigroup presentation $\mathcal{P}_{n,\ell}$ and the word $\omega_{n,\ell}$ can be computed from the $n$-ary expansions of the numbers $r_{i}$. If all entries in $\ell$ are rational, then $G_{n,\ell}$ has type $F_{\infty}$. Otherwise, $G_{n,\ell}$ is not finitely generated.