General Properties
- Property (T) is preserved under quotients: if G has property (T) and H is a quotient group of G then H has property (T). Equivalently, if a homomorphic image of a group G does not have property (T) then G itself does not have property (T).
- If G has property (T) then G/ is compact.
- Any countable discrete group with property (T) is finitely generated.
- An amenable group which has property (T) is necessarily compact. Amenability and property (T) are in a rough sense opposite: they make almost invariant vectors easy or hard to find.
- Kazhdan's theorem: If Γ is a lattice in a Lie group G then Γ has property (T) if and only if G has property (T). Thus for n ≥ 3, the special linear group SLn(Z) has property (T).
Read more about this topic: Kazhdan's Property (T)
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