A theorem (I do unfortunately not remember to whom it is due) states that there exists a finitely presented group containing a subgroup isomorphic to the additive group of rational numbers. Can somebody give an explicit construction?



An explicit example of a finitely presented group containing a subgroup isomorphic to $(\mathbb Q,+)$.
Francesco Matucci, James Hyde and I have just posted an [arXiv preprint](https://arxiv.org/abs/2005.02036) with a solution to this problem. We prove that $\mathbb{Q}$ embeds in the group $\overline{T}$ of piecewise-linear homeomorphisms of the real line obtained by lifting Thompson's group $T$ through the covering map from the line to the circle. That is, $\overline{T}$ consists of all piecewise-linear homeomorphisms $f$ of the real line that satisfy the following conditions:


1. Each linear segment of $f$ has the form $f(x) = 2^a x + \dfrac{b}{2^c}$ for some $a,b,c\in\mathbb{Z}$.
2. Each breakpoint of $f$ has dyadic rational coordinates.
3. $f(x+1)=f(x)+1$ for all $x\in\mathbb{R}$.


We also prove that this group $\overline{T}$ has a presentation with two generators and four relations.


It follows from this result together with a theorem of Brin that $\mathbb{Q}$ embeds in the automorphism group of Thompson's group $F$, which is also finitely presented. Similarly, $\mathbb{Q}$ embeds in the braided Thompson group $BV$ introduced by Brin and Dehornoy, which again is known to be finitely presented.


Our preprint also proves that $\mathbb{Q}$ embeds into a finitely presented simple group which we denote $T\mathcal{A}$. (None of the other groups listed above are simple.) This is a certain group of homeomorphisms of the circle that are "nearly piecewise-linear" in the sense that they have infinitely many linear pieces that accumulate at finitely many points. We prove that this group $T\mathcal{A}$ is two-generated and has type $\mathrm{F}\_\infty$, and we indicate how an explicit finite presentation of $T\mathcal{A}$ could be derived.


