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.)?