I have a set of two (not necessarily entirely related) questions about hyperbolic groups - I'll ask them as one question since they seem related enough and I have a feeling someone who knows the answer to one of the questions might know the answer to the other. Sorry for my ignorance on the subject - any pointers to references would be appreciated. OK, here are the questions:
(1) Is there an algorithm to determine if a given finitely-presented group is hyperbolic?
(2) Given a finitely-generated subgroup $H$ in a given finitely-presented group $G$ and a word $w$ in the generators of $G$, is there an algorithm to determine if $w \in H$? What about the same question but restricted to finite index $H$? For the case where $H=1$, I am aware of the solvability of the word problem for hyperbolic groups, but I don't know about the more general case.
Thanks for the help.
The answer to (1) is no, but it is possible to verify hyperbolicity (see the link in Dietrich Burde's comment).
(2) is no in general - this is called the Generalized Word Problem. The word problem in $G$ is equivalent to the case $H=1$. As you remarked, the word problem is solvable in hyperbolic groups, but there are examples in which the generalized word problem is not solvable.
Such examples can be constructed using a very simple technique due to Rips. Starting with an arbitrary finitely presented group $Q$, you can construct a finite presented group $G$ that satifies arbitrarily strong small-cancellation conditions, and which has a finitely generated normal subgroup $H$ with $G/H \cong Q$.
The answer is yes for subgroups that are known to be of finite index - you can do it by coset enumeration - but of course the problem of determining whether $|G:H|$ is finite is unsolvable.