Embedding $\mathbb{Q}$ into a Finitely Presented Group

Abstract

We prove that the lift of Thompson's group T to the real line contains the additive group Q of the rational numbers. This gives an explicit, natural example of a finitely presented group that contains Q, answering a Kourovka notebook question of Martin Bridson and Pierre de la Harpe.We also prove that Q can be embedded into a finitely presented simple group. Specifically, we describe a simple group TA of homeomorphisms of the circle that contains Q, and we prove that TA is two-generated and has type F∞. Our method for proving finiteness properties extends existing techniques to allow for groups of homeomorphisms that are locally more complicated than the standard Thompson groups.

The BNSR-invariants of the Lodha–Moore groups, and an exotic simple group of type

“…First note that since T is simple, it lies in the kernel of the abelianisation. Next, since the partial action of T on the set of non-empty finite binary sequences is transitive, using relations ( 5), (6) we conclude that y s and y t , s, t ∈ {0, 1} N \ {∅} are conjugate by elements of T. This implies that their images are equal. From relation (4) we also get that the image of y ∅ equals those of all the other y s .…”

“…These were Thompson's group T and its various generalisations due to Higman (see [15]) and Stein (see [21]). A novel new example appears in [6]. Moreover, in [14], Ghys and Sergiescu proved that T admits a faithful action by C ∞ -diffeomorphisms of the circle.…”