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Let $I$ be a finite category and $F \colon I \rightarrow \mathrm{Grp}$ be a functor into the category of groups, such that $F(c)$ is a finite group for every $c \in Obj(I)$.

Question. Does $\mathrm{colim}\ F$ always have a solvable word problem?

Is there in fact a uniform solution for the word problem (taking also $I$ and $F$ appropriately encoded as input)? I think we can wlog. impose that $F(f)$ is a monomorphism for every morphism $f$ in $I$, I would be also happy for an answer under this assumption.

A closely related construction is the fundamental group of a graph of groups, and there I think we do have a solvable word problem if all vertex groups are finite, as there is a nice normal form by Bass-Serre theory. Some (but not all!) colimits can be realized as the fundamental group of a graph of groups (I guess "on the nose" if $I$ is the path category of a forest and we only allow monomorphisms?)

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    $\begingroup$ Fundamental groups of graphs of groups are sort of homotopy colimits of groups, over categories whose nerve is a connected graph. If you allow yourself homotopy colimits of trivial groups over finite categories whose nerves are finite connected 2-complexes, then you can get any group presentation with finitely many generators and relations that you want. So there is definitely no higher dimensional version of your statement about fundamental groups of graphs of groups. These things are called `complexes of groups'. $\endgroup$ Commented Feb 18 at 17:35
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    $\begingroup$ If I'm correct: if we replace each group $F(c)$ with its image in the colimit, then we boil down to the case where each map is injective. (Still this is very far from Bass-Serre theory.) $\endgroup$ Commented Feb 18 at 17:45
  • $\begingroup$ I have a suspicion that this is exactly the same thing as studying complexes of finite groups where the underlying complex is simply connected. You should probably look at the final chapter of Bridson--Haefliger, where they develop the theory or complexes of groups using what they call "scwols" (Small Categories Without Loops). They prove a developability theorem: if a suitable version of Gromov's Link Condition is satisfied, then the resulting group is CAT(0), and in particular has solvable word problem. Without that condition, there's no reason to expect solvable word problem. $\endgroup$ Commented Feb 21 at 11:13

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