TITLE: Are there any computational problems in groups that are harder than P?
QUESTION [24 upvotes]: There are several well known classes of groups for which the word problem, conjugacy etc. are solvable in polynomial time (hyperbolic, automatic). 
Then there are several classes of groups like asynchronously automatic groups for which it is known that there is an exponential time algorithm to solve the word problem (and whether this can be improved to polynomial is open and conjectured as far as I'm aware).
Going several steps further, there is an algorithm to solve the word problem in one-relator groups in time not bounded by any finite tower of exponentials (and again it is open and conjectured whether this can be improved to P).
On the other, there are algorithms to solve word problems in pathological groups like the Baumslag-Gersten group:
$$G_{(1,2)} = \langle a, b | a^{a^b}= a^2 \rangle$$
in polynomial time. So even though general algorithms can be very bad, it is unknown whether there are group-specific algorithms for every group in a given class that solve the word problem quickly.
So my question is, are there any groups in which it is known that the word problem (or any other computational problem) is at least exponential, but still solvable? Maybe exp-complete? What are the approaches to proving such a thing?

REPLY [2 votes]: While most other answers have mentioned computational problems related to finitely presented (but generally infinite) groups, there are many problems in finite group theory which are either conjectured or known to be NP-complete. As an example, we have the following result from A. Lubiw, "Some NP-complete problems similar to graph isomorphism", SIAM J. Comp. 10(1981) 11-21.

The problem of determining whether $G$ has a fixed point free element is NP-complete, even for elementary abelian $2$-groups.

The classic problem of determining the number of subgroups of a group of order $n$ is also, as far as I can tell, conjectured to be very difficult, but I am not aware of any definite statement in either direction.
Moving back to the world of finitely presented groups, the subset sum problem is defined as follows. Given $g_1, \dots, g_k, g \in G$, decide whether $$ g = g_1^{\varepsilon_1} \cdots g_k^{\varepsilon_k}$$ for some $\varepsilon_1, \dots \varepsilon_k \in \{ 0, 1\}$. This problem is known to be NP-complete for a vast range of different classes of groups, including, but not limited to, free metabelian non-abelian groups of finite rank, and the wreath product of two finitely generated infinite abelian groups. It is also known to be NP-complete in some particular cases, such as for $BS(1, 2)$ and Thompson's group $F$. In hyperbolic groups, however, the problem is P-time decidable, so the problem certainly is an interesting one. 
The subset sum problem and many other related problems for finitely presented groups are studied in detail in A. Miasnikov, A. Nikolaev, A. Ushakov, "Knapsack Problems in Groups", Math. of Comp. 84(2013).