I have a few examples of a group $G$, equipped with a Hausdorff minimal nontrivial group topology $\cal T$. This means that $\cal T$ is Hausdorff and for any nontrivial (not necessarily Hausdorff) group topology $\cal S$ on $G$ with $\cal S\subseteq T$ we have $\cal S = T$. However, in these examples $\cal T$ is unique.


Is there an example of a group $G$ such that there are more than one Hausdorff minimal nontrivial group topology on $G$?



A group with more than one Hausdorff minimal nontrivial group topologies
Yes, many.


Here is an example.
Take two different locally compact groups with minimal Hausdorff topologies, say $G\_1=\text{PSL}\_2(\mathbb{R})$ and $G\_2=\text{PSL}\_2(\mathbb{Q}\_p)$
for some prime $p$ (endowed with their usual topologies, which are minimal by Theorem 5.3 of <http://arxiv.org/pdf/1408.4217.pdf>),
and take a group $\Gamma$ that injects densely into both, say $\Gamma=\text{PSL}\_2(\mathbb{Q})$ (you could also take $\Gamma$ to be a free group here).
Take $T\_i$ on $\Gamma$ to be the pull back of the topology of $G\_i$.


Note that for each $i=1,2$, the group $G\_i$ is the completion (wtr the two sided uniform structure) of $(\Gamma,T\_i)$, by the fact that $\Gamma$ is dense and $G\_i$ is complete (as it is locally compact). 
In particular, $T\_1\neq T\_2$.


We are left to show that the topologies $T\_1$ and $T\_2$ are minimal.
Assume that for some $i\in\{1,2\}$, $S\lneq T\_i$ is a weaker topology.
Let $H$ be the completion of $(\Gamma,S)$. The map $\Gamma\to H$ extends canonically to $G\_i\to H$ and we get by minimality of $G\_i$ that $H$ is trivial.
Thus $S$ is trivial.


As a reply to a comment below, note that the groups $G\_i$ are necessarily simple: otherwise they would have a weaker (non-Hausdorff) topologies. 


