Suppose $G$ is a discrete group and $H \leq G$ a subgroup of finite index. If $H$ has Kazhdan property (T), does it follow that $G$ has property (T)? (I've read somewhere that (T) is preserved by exact sequences, so if $N$ is normal and $G/N$ is finite, then the fact above holds ; here, however, we do not assume $H$ to be normal)



Property (T) and subgroups of finite index
Yes. For groups $H\subset G$, with H a lattice, H has (T) iff G has (T). When both groups are discrete being a lattice is the same as being finite index.


Almost every thing you ever need to know about Property (T) can be found here


<http://perso.univ-rennes1.fr/bachir.bekka/KazhdanTotal.pdf>


