  We 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.
