What is an example of  a torsion free discrete abelian group $G$ whose dual space $\hat{G}$ is not a path connected space?


**The Motivation:**  The motivation comes from the idempotent conjecture of Kaplanski: For a torsion free and discreet abelian group $G$ the dual group $\hat{G}$ is  a connected space so $C^*_{red} G \sim C(\hat{G})$ has no nontrivial idempotent. So in this case the  Kaplanski conjecture is true. So we wish to reduce the power of connectivity as much as possible. For example we would like to think to a possible discrete group $G$ which is torsion free and some of its [reasonable dual](https://link.springer.com/article/10.1134/S0081543810040164#author-information) or other [types of dual](https://en.wikipedia.org/wiki/Pontryagin_duality#Dualities_for_non-commutative_topological_groups) or [this one](https://mathoverflow.net/a/70658/36688)  would be  weak connected as much as possible.