There is a famous open problem in group theory that asks:

Does there exist an infinite finitely presented torsion group?

The general belief being that such groups exist. I would like to know whether the existence of such a group can be ruled out if we impose stronger finiteness conditions, i.e.

Is any torsion group $G$ with a $K(G,1)$ of finite type necessarily finite?

Here, finite type means that the CW-complex $K(G,1)$ has finitely many cells in any dimension. In the previous question, one could impose even stronger conditions, e.g. requiring additionally for $G$ to have finite exponent.

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    $\begingroup$ The condition of having a $K(G,1)$ of finite type is called "of type $F_\infty$", and also equivalent to "finitely presented of type $FP_\infty$", where $FP_\infty$ is defined homologically. Reference: Brown's book on Cohomology of groups $\endgroup$ – YCor May 17 '16 at 8:47
  • $\begingroup$ I think proving or disproving this is well beyond the current technique. $\endgroup$ – Misha May 21 '16 at 15:58

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