In the book Geometric Group Theory of Clara Loh, it is proven that the Word Problem is solvable for hyperbolic groups. It is also stated that the Word Problem is not solvable in all finitely presented groups.
Now I was searching for an example of such a group with finite presentation for which the Word Problem is not solvable, but I can't find one.
Can anyone help me please? Thanks in advance!
As Derek mentioned, it's not easy, it was really hard to implement.
Nevertheless, let me somewhat mention relevant things:
It can be shown that $[x,t^ixt^{-i}]$ holds in $G$ if and only if $i\in I$. Hence, if $I$ is not computable, then $G_I$ doesn't have a solvable word problem.
It is obvious that if $I$ is recursively enumerable then $G_I$ is recursively presented. So if $I$ is recursively enumerable and not computable (i.e. the complement of $I$ is not recursively enumerable) then $G_I$ is recursively presented without solvable word problem.
Now a theorem of Higman is that every finitely generated recursively enumerable group embeds into a finitely presented group. If we take this as granted (this is hard!) we get a finitely presented group without solvable word problem.
Of course this Higman embedding theorem is a black box, but still this gives an indication what kind of phenomena can prevent a finitely presented group to be computable (=to have solvable word problem).
I should also mention that most results showing that some kind of computability problems is not solvable, consist in "encoding" solvability of the halting problem within the given kind of structure. For f.p. semigroups, it isn't that hard (to then produce non-computable f.p. semigroups, i.e., with non-solvable equality problem). For groups it's technically much more involved.