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

Abstract

Assume $n\ge 2$ and $\ell =(r_1,\dots ,r_k)\in [0,1{]}^k$ 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) $h^{\prime }(x)$ is a power of $n$ where it is defined; (ii) if $h^{\prime }(x)$ 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 $n\ge 2$ 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.

27 pages, 4 figures. Updated with a new section on alternate approaches to the main theorem and overlap with work of others