A short account of why Thompson's group $F$ is of type ${\text{F}}_{\infty }$

Abstract

In 1984 Brown and Geoghegan proved that Thompson's group $F$ is of type ${\text{F}}_{\infty }$, making it the first example of an infinite dimensional torsion-free group of type ${\text{F}}_{\infty }$. Over the decades a different, shorter proof has emerged, which is more streamlined and generalizable to other groups. It is difficult, however, to isolate this proof in the literature just for $F$ itself, with no complicated generalizations considered and no additional properties proved. The goal of this expository note then is to present the "modern" proof that $F$ is of type ${\text{F}}_{\infty }$, and nothing else.

Expository note. 8 pages, 3 figures. v2: Corrected the reference [BG84]. v3: Incorporated referee's suggestions (including title change). Accepted version, to appear in Topology Proc