An example for a non-precompact minimal topological group. Do you have an example of a non-precompact minimal topological group?
A topological group $(G,\mathcal T)$ is said to be minimal iff it is Hausdorff and for any compatible Hausdorff topology $\mathcal S$ on $G$, we have:
$$\mathcal S\subseteq\mathcal T\to\mathcal S=\mathcal T$$
I think Ivan Prodanov proved all abelian minimal topological groups are precompact. So the example needs to be non-abelian.
precompactness is defined with left topological group uniformity.
 A: Yes, there was an article “Every minimal abelian group is precompact” [PS] by Ivan Prodanov and Luchesar Stojanov. About 15 years ago my teacher gave me this paper. As I remember, he told to me Prodanov and Stojanov searched for the proof for ten years.  
A simple example of a non-precompact minimal topological group should be ${\bf SL}(2,\mathbb R)$ (see part 7 and, especially, section 7.4 of the book [DPS] by Dikran Dikranjan, Ivan Prodanov, and Luchesar Stojanov). Also A.I. Costash, and Mikhail Ursul in [CU] constructed examples of non-compact locally compact minimal groups. 
Some of our investigation is closely related with minimal topological groups. So, if you have any questions about them, feel free to ask. :-) 
References
[CU] A.I. Costash, M.I. Ursul. Examples of minimal topological groups
[DS] Dikranjan D., Prodanov I., Stoyanov L. Topological Groups. Marcel Dekker, New-York, 1990
[PS] I. Prodanov, L.N. Stojanov, Every minimal abelian group is precompact, C. R. Acad. Bulgar. Sci. – 1984. – 37, #1.
Here they are: paper [CU], book [DS], and the first page of the paper [PS], which my teacher gave me. :-) 
