We prove that, for a finitely generated residually finite group, having solvable word problem is not a sufficient condition to be a subgroup of a finitely presented residually finite group. The obstruction is given by a residually finite group with solvable word problem for which there is no effective method that allows, given some non-identity element, to find a morphism onto a finite group in which this element has a non-trivial image. We also prove that the depth function of this group grows faster than any recursive function.

We systematically study groups whose marked finite quotients form a recursive set. We give several definitions, and prove basic properties of this class of groups, and in particular emphasize the link between the growth of the depth function and solvability of the word problem. We give examples of infinitely presented groups whose finite quotients can be effectively enumerated. Finally, our main result is that a residually finite group can fail to be recursively presented and still have computable finite quotients, and that, on the other hand, it can have solvable word problem but not have computable finite quotients.

We make the case that the so called "global decision problems" should not be investigated solely for groups described by finite presentations. We propose to use descriptions that be algorithms that perform some given tasks, and that encode the considered groups. We motivate this by establishing undecidability results for groups described by recursive presentations, strong enough to prevent an interesting theory of decision problems based on generic recursive presentations to be developed. More importantly, we give an algorithmic characterization of finitely presented groups, in terms of existence of a "marked quotient algorithm" which recognizes the quotients of the considered group. This new point of view leads us to proposing several open questions and directions of research, and much of this paper consists in exposing problems that arise from our first results. Finally, note that we set our study in the category of marked groups, we explain why this is beneficial, and give open questions that arise from the study of decision problems for marked groups.