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I am a beginner in Kazhdan's property (T). So, my questions may look very elementary. But I have not seen anything about them in the literature yet.

My first question: Given a group $G$, is there always a noncompact topology on $G$ under which $G$ has property (T)?

Assume $\tau_1 $ and $\tau_2$ are two different topologies on a group $G$ which make it a topological group. If it helps you can assume both are hausdorff. We also assume under none of these topologies $G$ is compact (because when $G$ is compact, it has property (T)!).

My second question: Assume $\tau_1 $ is weaker than $\tau_2$. If $(G,\tau_2)$ has (T), does it imply that $(G,\tau_1)$ has it too? What about vice versa?

My third question: Is there any topological condition (besides compactness) which is relevant to property (T) (for example being totally disconnected, $\sigma$-compact, locally compact, metrizable, etc.)?

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    $\begingroup$ Both parts of the second question are straightforward exercises. The third question is very vague, as if I were asking for topological spaces "is compactness relevant to connectedness"? The first question makes sense, but there are uncountable group with no other topology than the discrete and the undiscrete one, so the answer is no (since discrete uncountable groups aren't T). $\endgroup$
    – YCor
    Jan 15, 2014 at 17:51
  • $\begingroup$ @YvesCornulier: About the first question, is it a good idea to look for certain (algebraic or geometric) conditions on $G$, which guarantee the existence of a noncompact topology on $G$ under which $G$ has property (T)? $\endgroup$
    – user23860
    Jan 15, 2014 at 18:03

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Infinitely generated discrete groups do not have property (T). So if one takes a Lie group with property (T) (with the natural topology), the group with the discrete topology does not have (T), since it is infinitely generated.

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