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)
Famous quotes containing the words general and/or properties:
“The conclusion suggested by these arguments might be called the paradox of theorizing. It asserts that if the terms and the general principles of a scientific theory serve their purpose, i. e., if they establish the definite connections among observable phenomena, then they can be dispensed with since any chain of laws and interpretive statements establishing such a connection should then be replaceable by a law which directly links observational antecedents to observational consequents.”
—C.G. (Carl Gustav)
“The reason why men enter into society, is the preservation of their property; and the end why they choose and authorize a legislative, is, that there may be laws made, and rules set, as guards and fences to the properties of all the members of the society: to limit the power, and moderate the dominion, of every part and member of the society.”
—John Locke (1632–1704)