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

Assume $n\ge 2$ and $\mathrm{\ell }=(r_1,\dots ,r_k)\in [0,1{]}^k$ is an increasing sequence of real numbers. Let $G_{n,\mathrm{\ell }}$ denote the group of orientation-preserving piecewise linear homeomorphisms $h$ of $I=[r_1,r_k]$ such that: (i) $h^{\mathrm{\prime }}(x)$ is a power of $n$ where it is defined; (ii) if $h^{\mathrm{\prime }}(x)$ is undefined, then $x$ is an $n$-adic rational number, (iii) $h$ fixes each entry of $\mathrm{\ell }$, and (iv) $h(\mathbb{Z}[1\mathrm{/}n]\cap I)=\mathbb{Z}[1\mathrm{/}n]\cap I$. We prove that $G_{n,\mathrm{\ell }}$ is a diagram group $D({\mathcal{P}}_{n,\mathrm{\ell }},{\omega }_{n,\mathrm{\ell }})$ for all integers $n\ge 2$ and for all finite sequences $\mathrm{\ell }$. The semigroup presentation ${\mathcal{P}}_{n,\mathrm{\ell }}$ and the word ${\omega }_{n,\mathrm{\ell }}$ can be computed from the $n$-ary expansions of the numbers $r_i$. If all entries in $\mathrm{\ell }$ are rational, then $G_{n,\mathrm{\ell }}$ has type $F_{\mathrm{\infty }}$. Otherwise, $G_{n,\mathrm{\ell }}$ is not finitely generated.