In this talk, we describe a process whereby we can embed any
hyperbolic group into a finitely presented infinite simple group.  This
gives a proof of what is, in some sense, the ​``typical'' case of the
Boone-Higman Conjecture of 1973 (that a finitely generated group G has a
solvable word problem if and only if G can be embedded into a finitely
presented simple group). [1]

    The work features a blend of techniques from groups, dynamics, and
from automata theory, but we hope to support more general audiences in
the presentation.

  Joint with James Belk, Francesco Matucci, and Matthew Zaremsky. [1]