A universal group of type F-infinity containing every finitely presented group (F-infinity Higman embedding)

Theorem 1.1 constructs one group $H$ of type $F_\infty$ containing every finitely presented group; by Higman's theorem its finitely generated subgroups are exactly the finitely generated recursively presented groups (Corollary 1.2). No word-problem hypothesis is imposed. The classifying space need not be finite-dimensional, so type $F$ is not claimed, and $H$ is not claimed simple.

Theorem 1.1 constructs one group $H$ of type $F_\infty$ containing every finitely presented group; by Higman's theorem its finitely generated subgroups are exactly the finitely generated recursively presented groups (Corollary 1.2). No word-problem hypothesis is imposed. The classifying space need not be finite-dimensional, so type $F$ is not claimed, and $H$ is not claimed simple.