Suppose $G$ is a finitely generated group, $A$ is a finite set of generators of $G$. Define $\pi: (A \cup A^{-1})^* \to G$ using the following recurrence:
$$\pi(\Lambda) = e$$ $$\pi(a \alpha) = a\pi(\alpha)$$
Now define the language $L(G, A) := \{w \in (A \cup A^{-1})^*| \pi(w) = e\}$.
It seems, that position of $L$ in Chomsky hierarchy uniquely depends on some structural properties of $G$. There are the following three theorems about that:
Anisimov theorem
$L(A, G)$ is regular (type 3 in Chomsky hierarchy) iff $G$ is finite.
Muller-Shupp theorem
$L(A, G)$ is context-free (type 2 in Chomsky hierarchy) iff $G$ is virtually free.
Higman theorem
$L(A, G)$ is recursively enumerable (type 0 in Chomsky hierarchy) iff $G$ is isomorphic to a subgroup of a finitely presented group.
You may have already noticed that something (and, to be exact, context-sensitive languages - type 1 in Chomsky hierarchy) is missing from the list. My question is:
Can the condition of context-sensitivity of $L(G, A)$ also be equivalently characterised through some group-theoretic properties of $G$.