TITLE: Empty preimage under homomorphism of finitely presented groups with decidable word problems
QUESTION [5 upvotes]: Let $f:G\to H$ be a homomorphism of finitely presented groups with decidable word problems.
Assume you are given explicit finite presentations for both $G$ and $H$ and you are given the words to which $f$ sends the generators of $G$.

Question. Is it possible that no algorithm exists deciding, given $w\in H$, whether $f^{-1}(w)$ is empty or not?

P.S. Having explicit finite presentations is important https://arxiv.org/abs/1003.5117

REPLY [5 votes]: It is known that there exists a finitely presented group $H$ with solvable word problem that has a finitely generated subgroup $K$ whose subgroup membership problem is unsolvable. For example, Mikhailova has shown that the product $F_2\times F_2$ of non-abelian free groups has such a subgroup (see here).
Now let $k_1,\ldots,k_n$ be the generators for $K$, let $G$ be a free group with $n$ generators $x_1,\ldots,x_n$, and let $f\colon G\to H$ be the homomorphism that maps each $x_i$ to $k_i$.  Then a given element $h\in H$ has the property that $f^{-1}(h)$ is nonempty if and only if $h\in K$, and therefore it is undecidable in general whether $f^{-1}(h)$ is nonempty.