the groups $f \leq t \leq v$ were defined by richard thompson in 1965 and used to construct finitely presented groups with unsolvable word problems. $t$ and $v$ were also the first examples of infinite, finitely presented simple groups. since then, these groups have been studied extensively using a rich interplay of algebraic, topological, and dynamical approaches. i will discuss recent work regarding the higher dimensional analogues of thompson groups, $nv$, including the fact that $mv$ is not isomorphic to $nv$ for $n \neq m$, and that for every $n$ the group $nv$ is finitely presented and simple. the only background required for this talk is basic group theory.