  A trisection of a smooth, closed, oriented 4-manifold is a decomposition into
three 4-dimensional 1-handlebodies meeting pairwise in 3-dimensional
1-handlebodies, with triple intersection a closed surface. The fundamental
groups of the surface, the 3-dimensional handlebodies, the 4-dimensional
handlebodies, and the closed 4-manifold, with homomorphisms between them
induced by inclusion, form a commutative diagram of epimorphisms, which we call
a trisection of the 4-manifold group. A trisected 4-manifold thus gives a
trisected group; here we show that every trisected group uniquely determines a
trisected 4-manifold. Together with Gay and Kirby's existence and uniqueness
theorem for 4-manifold trisections, this gives a bijection from group
trisections modulo isomorphism and a certain stabilization operation to smooth,
closed, connected, oriented 4-manifolds modulo diffeomorphism. As a
consequence, smooth 4-manifold topology is, in principle, entirely group
theoretic. For example, the smooth 4-dimensional Poincar\'e conjecture can be
reformulated as a purely group theoretic statement.
