Reference for: the fundamental group of a simplicial complexes is in general undecidable

Question

For my thesis I would like to cite the fact that there is a simplicial complex whose fundamental group is undecidable. However, I can not find a reference (or a cluster of references) which explicitly state so.

Alternatively I could also re-prove this fact using the Theorem of Novikov-Boone. I know how to show the existence of CW-complexes with undecidable fundamental groups, using the Theorem of Novikov-Boone. This is fairly easy. But for me it requires quite some work to achieve the same for simplicial complexes.

For some perspective: my focus is on logic and discrete mathematics. I have very basic knowledge of algebraic geometry. I also have to stick to the setting of simplicial complexes.

Answer 1

Regardless of properties of $G$: if you can construct a CW complex that has $G$ as a fundamental group then you can also construct such simplicial complex. It follows from the following theorem:

Theorem: Every CW complex X is homotopy equivalent to a simplicial complex, which can be chosen to be of the same dimension as X , finite if X is finite, and countable if X is countable.

The theorem with a proof can be found in Allen Hatcher's "Algebraic Topology", Theorem 2C.5, page 182. I recommend reading whole Hatcher if you are new to algebraic topology.