Embedding in f.p. simple groups Dear All!
At the time when Lyndon and Schupp wrote their book there was an open question:
Question: Does every finitely presented group with soluble word problem embed in a finitely presented simple group?
Is it still open? Could you hint at some useful references about this? Thanks!
 A: I believe it is still open. By the Boone-Higman Theorem (W. W. Boone and G. Higman, "An algebraic characterization of the solvability of the word problem", J. Austral. Math. Soc. 18, 41-53 (1974)), a finitely presented group has solvable word problem if and only if it can be embedded in a simple group that can be embedded in a finitely presented group.
It is widely believed that it is possible for the simple group itself to be finitely presented, but (AFAIK) not proved.
So the answer to Agol's comment is that no finitely presented group with unsolvable word problem can be embedded into a finitely presented simple group.
A: There is a strengthening of the Boone-Higman result, due to Thompson. He showed that we can take the simple group to be finitely generated. In full, this reads:
"A finitely presented group has solvable word problem if and only if it can be embedded in a finitely generated simple group that can be embedded in a finitely presented group."
You can find the full details in:
R. J. Thompson, "Embeddings into finitely generated simple groups which preserve the word problem", Word Problems II: The Oxford Book, Studies in Logic and the Foundations of Mathematics, Volume 95, (1980).
As far as I am aware, your original question "Does every finitely presented group with soluble word problem embed in a finitely presented simple group?" is still an open problem.
-Maurice
