There is a natural notion of a presentations in the category of residually finite groups. Namely, if $X$ is set and $R$ is a set of words in the free group $FG(X)$ on $X$, then define $G=RF\langle X\mid R\rangle$ to be $FG(X)/N$ where $N$ is the intersection of all *finite index* normal subgroups of $FG(X)$ containing $R$. Equivalently, $N$ is the closure in the profinite topology of the normal closure of $R$. The group $G$ is residually finite and is the universal residually finite quotient of the group $H$ with presentation $\langle X\mid R\rangle$ (equivalently, it is the quotient of $H$ by the closure of $\{1\}$ in the profinite topology).

Every residually finite group has a presentation in this sense.

Having a finite presentation in the residually finite category does not (or at least should not) imply being finitely presented in the usual sense.

One can then ask about the uniform word problem for residually finite groups in this setting. That is, we can ask given a finite set $X$, a finite set of relations $R\subseteq FG(X)$ and a word $w\in FG(X)$, is $w=1$ in $G=RF\langle X\mid R\rangle$? Notice that the set of such $w$ is co-r.e. because we can enumerate all $X$-generated finite groups satisfying $R$ and determine if $w\neq 1$ in some such finite group. But there is no procedure to enumerate the words $w$ equal to $1$ in $G$. Indeed, Slobodoskoii proved the uniform word problem is undecidable for residually finite groups.

One can then ask the following restricted problem.

Question.Is the uniform word problem decidable for groups with a one-relator residual finite presentation? That is, given a finite set $X$ and a word $r\in FG(X)$, is there an algorithm to determine if a word $w\in FG(X)$ is trivial in $G=RF\langle X\mid R\rangle$?

This is basically asking for a Magnus theorem in the category of residually finite groups. Not all $1$-relator groups (in the usual sense) are residually finite so being $1$-relator as a residually finite group does not (*a priori*) mean being a $1$-relator group.

My question is equivalent to asking whether there is an algorithm to compute membership in the profinite closure of the trivial subgroup of a $1$-relator group.