Geometric presentations for Thompson's groups and the multiscaled braid group
Résumé
This paper deals with Thompson's groups~$F$ and~$V$, and two related groups, namely a subgroup~$V'$ of~$V$ and an extension~$F\!B_\infty$ of~$V'$ connected with the latter as Artin's braid group~$B_\infty$ is connected with the symmetric group~$\Sym_\infty$. The latter group also appears (under the name $\widehat{BV}$) in the independent work~\cite{Bri}. Our aim in this text is to investigate these groups from a geometric point of view relying on their connection with the associativity and commutativity laws. This approach leads in particular to new presentations. Using word reversing, a specific combinatorial method, we can derive a number of algebraic properties from the presentations. We prove that the group~ $F\!B_\infty$ is torsion-free and even orderable, that it includes (and is generated by) both Thompson's group~$F$ and Artin's braid group~$B_\infty$ (whence our notation). Also it can be interpreted as a group of braids involving a fractal sequence of strands, and it is connected with a twisted version of commutativity and the self-distributivity law.