New Presentations of Thompson's Groups and Applications

We find new presentations for the Thompson’s groups $F$, the derived group $F^{{}^{\mathrm{\prime }}}$ and the intermediate group $D$. These presentations have a common ground in that their relators are the same and only the generating sets differ. As an application of these presentations we extract the following consequences: the cost of the group $F^{{}^{\mathrm{\prime }}}$ is 1 hence the cost cannot decide the (non)amenability question of $F$; the $I{I}_1$ factor $L(F^{{}^{\mathrm{\prime }}})$ is inner asymptotically abelian and the reduced $C^{\ast }$-algebra of $F$ is not residually finite dimensional.