show/hide this revision's text 2 Fixed some typos

I hope I have not goofed, but I think the answer to your modified question is yes:

The fundamental group of a semi-locally simply connected, compact and geodesic space is finitely presented.

Here are the ingredients - all numbers in parentheses refer to Bridson-Haefliger, Metric spaces of non-positive curvature, Springer Grundlehren, 1999, Part I.

  • The universal covering space $\widetilde{X}$ equipped with the length metric induced by the covering projection is a length space (3.25).
  • The fundamental group $\pi_{1}(X)$ acts on $\widetilde{X}$ properly and cocompactly by isometries (8.3 (2)), see also (8.5).
  • If a length space $\tilde{X}$ \widetilde{X}$ admits a proper and cocompact action by isometries then it is locally compact (8.4 (1)) and hence proper and geodesic (3.7).
  • A group is finitely presented if and only if it acts properly and cocompactly by isometries on a simply connected geodesic space (8.11).

All this taken together yields that $\widetilde{X}$ is a simply connected geodesic metric space and $\pi_{1}{(X)}$ acts properly and cocompactly by isometries, hence $\pi_{1}{(X)}$ must be finitely presented.
show/hide this revision's text 1

I hope I have not goofed, but I think the answer to your modified question is yes:

The fundamental group of a semi-locally simply connected, compact and geodesic space is finitely presented.

Here are the ingredients - all numbers in parentheses refer to Bridson-Haefliger, Metric spaces of non-positive curvature, Springer Grundlehren, 1999, Part I.

  • The universal covering space $\widetilde{X}$ equipped with the length metric induced by the covering projection is a length space (3.25).
  • The fundamental group $\pi_{1}(X)$ acts on $\widetilde{X}$ properly and cocompactly by isometries (8.3 (2)), see also (8.5).
  • If a length space $\tilde{X}$ admits a proper and cocompact action by isometries then it is locally compact (8.4 (1)) and hence proper and geodesic (3.7).
  • A group is finitely presented if and only if acts properly and cocompactly by isometries on a simply connected geodesic space (8.11).

All this taken together yields that $\widetilde{X}$ is a simply connected geodesic metric space and $\pi_{1}{(X)}$ acts properly and cocompactly by isometries, hence $\pi_{1}{(X)}$ must be finitely presented.