A group with more than one Hausdorff minimal nontrivial group topologies

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 $G$?

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.

• It would be nice if you could give a reference for the fact that the $G_i$ are minimal. Also, in the last step one should require that the $T_i$ are essential in $G_i$ (see caminos.udc.es/gmni/pdf/2011/locally_minimal_groups_2.pdf Theorem 3.1), so possibly some free $\Gamma$ might not be essential (?). – Guntram Sep 8 '16 at 15:52
• @Guntram, thanks for your comment. I will add more details later, when I have more than a few minutes. This is just to address your criticism: note that $G_i$ are simple, so every dense sg is essential. – Uri Bader Sep 8 '16 at 15:59
• @Guntram reference and a few more details were added. – Uri Bader Sep 9 '16 at 7:52