2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/98690We prove that Thompson's groups $F$ and $V$ are the geometry groups of associativity, and of associativity together with commutativity, respectively. We deduce new presentations of $F$ and $V$. These presentations lead to considering a certain subgroup of $V$ and an extension of this subgroup. We prove that the latter are the geometry groups of associativity together with the law $x(yz) = y(xz)$, and of associativity together with a twisted version of this law involving self-distributivity, respectively.Group TheoryMSC: 20F05, 20F36, 20B07text