Do you have an example of an infinite simple group with at least 3 distinct group topologies on it?
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2$\begingroup$ $PSL_2(\mathbf{Q})$. It has infintely many group topologies, namely those induced by the embeddings $\mathbf{Q}\subset\mathbf{Q}_p$ where $p$ ranges over primes (possibly including $\mathbf{Q}_0=\mathbf{R}$ as well). $\endgroup$– YCorCommented May 23, 2014 at 11:26
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$\begingroup$ Since the discrete and the indiscrete topology are always compatible with the group structure, having three topologies isn't really that big of a requirement. In particular: Any (at least one-dimensional) group theoretically simple Lie-group will satisfy it. $\endgroup$– Johannes HahnCommented May 23, 2014 at 14:09
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$\begingroup$ @JohannesHahn: I didn't have an example of topologizable but Alexandrov non-topologizable. I'm not so familiar with Lie groups and matrix groups. $\endgroup$– Minimus HeximusCommented May 23, 2014 at 14:39
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3$\begingroup$ @MinimusHeximus You don't have to be "familiar". That real matrix groups are topological groups w.r.t. the euclidean topology is immediately obvious. $\endgroup$– Johannes HahnCommented May 23, 2014 at 14:49
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Here is a suggestion take $SL_2(\mathbb{Q}_p)$. Then it has the topology as a matrix group over $\mathbb{Q}_p$. It has also the discrete topology. Finally you can embed $\mathbb{Q}_p$ in $\mathbb{C}$ (in many ways). So it has many topologies as a matrix group over $\mathbb{C}$. Now, you can divide by the centre and get $PSL(\mathbb{Q}_p)$. But I am not 100% sure that all of these topologies are distinct.
answered May 23, 2014 at 11:18