The Boone-Higman conjecture (1973) predicts that a finitely generated group has solvable word problem if and only if it embeds in a finitely presented simple group. The "if" direction is true and easy, but the "only if" direction has been open for over 50 years. In this talk I will discuss an interesting sufficient condition for a group to satisfy the conjecture, which does not require one to care about simple groups, but rather to care about group actions with certain properties. This viewpoint has led to a number of recent breakthroughs on the conjecture, such as proving it for all hyperbolic groups, and for the groups $Aut(F_n)$. In all these cases the simple groups that arise are certain groups of Cantor space homeomorphisms with interesting dynamics, called "twisted Brin-Thompson groups". This talk will touch on joint work with combinations of Jim Belk, Collin Bleak, Francesco Fournier-Facio, James Hyde, and Francesco Matucci.