Property (T) and subgroups of finite index - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-22T21:46:01Z http://mathoverflow.net/feeds/question/43537 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/43537/property-t-and-subgroups-of-finite-index Property (T) and subgroups of finite index Marcin Kotowski 2010-10-25T16:09:42Z 2010-10-25T22:25:00Z <p>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)</p> http://mathoverflow.net/questions/43537/property-t-and-subgroups-of-finite-index/43540#43540 Answer by Owen Sizemore for Property (T) and subgroups of finite index Owen Sizemore 2010-10-25T16:31:20Z 2010-10-25T16:31:20Z <p>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.</p> <p>Almost every thing you ever need to know about Property (T) can be found here</p> <p><a href="http://perso.univ-rennes1.fr/bachir.bekka/KazhdanTotal.pdf" rel="nofollow">http://perso.univ-rennes1.fr/bachir.bekka/KazhdanTotal.pdf</a></p> http://mathoverflow.net/questions/43537/property-t-and-subgroups-of-finite-index/43541#43541 Answer by Jon Bannon for Property (T) and subgroups of finite index Jon Bannon 2010-10-25T16:32:19Z 2010-10-25T19:38:53Z <p>I think even more is true. See Proposition 2.5.5 in "the book":</p> <p>Bekka, Bachir; de la Harpe, Pierre; Valette, Alain (2008), Kazhdan's property (T), New Mathematical Monographs, 11, Cambridge University Press, MR2415834, ISBN 978-0-521-88720-5, <a href="http://perso.univ-rennes1.fr/bachir.bekka/KazhdanTotal.pdf" rel="nofollow">http://perso.univ-rennes1.fr/bachir.bekka/KazhdanTotal.pdf</a> </p> <p>for (I think) a stronger result about property (FH)</p> http://mathoverflow.net/questions/43537/property-t-and-subgroups-of-finite-index/43587#43587 Answer by Hadi for Property (T) and subgroups of finite index Hadi 2010-10-25T22:25:00Z 2010-10-25T22:25:00Z <p>Since $H$ contains a subgroup of finite index which is normal in $G$, what you are asking also follows from Theorem 1.7.1 of the book of Bekka, et. al. (mentioned in Jon Bannon's comment.) </p>