14.10 (1999)

a) (Well-known problem). It is known that any recursively presented group embeds in a finitely presented group (G. Higman, Proc. Royal Soc. London Ser. A, 262 (1961), 455–475). Find an explicit and “natural” finitely presented group $\Gamma$ and an embedding of the additive group of the rationals $\mathbb{Q}$ in $\Gamma$.
b) Find an explicit embedding of $\mathbb{Q}$ in a finitely generated group; such a group exists by Theorem IV in (G. Higman, B. H. Neumann, H. Neumann, J. London Math. Soc., 24 (1949), 247–254).
c) Find an explicit and “natural” finitely presented group $\Gamma_n$ and an embedding of $GL_n(\mathbb{Q})$ in $\Gamma_n$.

Another phrasing of the same problems is: find a simplicial complex $X$ which covers a finite complex such that the fundamental group of $X$ is $\mathbb{Q}$ or, respectively, $GL_n(\mathbb{Q})$.