Property T

In mathematics , property T (also Kazhdan's property T ) is a rigidity property of topological groups that was first considered by David Kazhdan in the 1960s.

Later developments showed that property T plays a role in many areas of mathematics, including discrete subsets of Lie groups , ergodic theory , random walks , operator algebras , combinatorics, and theoretical computer science .

A version that is used in evidence in the Zimmer program , among other things , is the strong property T introduced by Vincent Lafforgue .

definition

Let be a strongly continuous, unitary action of a topological group on a Hilbert space . ${\displaystyle \pi \colon G\to U({\mathcal {H}})}$ ${\displaystyle G}$ ${\displaystyle {\mathcal {H}}}$

For a compact set and a vector is called -invariant if ${\displaystyle Q\subset G}$ ${\displaystyle \epsilon >0}$ ${\displaystyle z\in {\mathcal {H}}}$ ${\displaystyle (Q,\epsilon )}$

${\displaystyle \sup _{g\in Q}\parallel \pi (g)z-z\parallel <\epsilon \parallel z\parallel }$.

${\displaystyle G}$has property T if there is a compact set and a such that for every unitary action there is an -invariant vector. ${\displaystyle Q\subset G}$ ${\displaystyle \epsilon >0}$ ${\displaystyle (Q,\epsilon )}$

Examples

• Every compact group has property T. You can choose and . ${\displaystyle Q=G}$ ${\displaystyle \epsilon ={\sqrt {2}}}$
• ${\displaystyle \mathbb {R} ^{n}}$and do not have property T. ${\displaystyle \mathbb {Z} ^{n}}$
• A locally compact group is compact if and only if it is indirect and has property T.
• ${\displaystyle SL(n,\mathbb {R} )}$has property T if and only if is. More generally, for each local body, the groups with and with property T. ${\displaystyle n\geq 3}$ ${\displaystyle K}$ ${\displaystyle SL(n,K)}$ ${\displaystyle n\geq 3}$ ${\displaystyle Sp(2n,K)}$ ${\displaystyle n\geq 2}$
• Simple Lie groups with have property T. ${\displaystyle rk_{\mathbb {R} }(G)\geq 2}$

properties

• Every locally compact group with property T is generated compactly. In particular, grids with property T are finite .
• If has property T, then has property T for every normal divisor . ${\displaystyle G}$ ${\displaystyle G/N}$ ${\displaystyle N}$
• If locally compact, closed and has a finite, regular , - invariant Borel measure , then has property T if and only if this applies to. In particular, a grid has property T if and only if this applies to. ${\displaystyle G}$ ${\displaystyle H\subset G}$ ${\displaystyle G/H}$ ${\displaystyle G}$ ${\displaystyle H}$ ${\displaystyle G}$ ${\displaystyle \Gamma \subset G}$ ${\displaystyle G}$
• According to Delorme-Guichardet's theorem , a group has property T if and only if it has property FH: every continuous action by affine isometries on a Hilbert space has a fixed point . Must be equivalent for all unitary representations . ${\displaystyle G}$ ${\displaystyle {\mathcal {H}}}$ ${\displaystyle H^{1}(G,\pi )=0}$ ${\displaystyle \pi \colon G\to U({\mathcal {H}})}$
• From property FH it follows, for example, that every effect by isometrics of a tree or a hyperbolic space must have a fixed point, or that every orientation- preserving effect of the group on the circle factors via the effect of a finite cyclic group . ${\displaystyle C^{\infty }}$

literature

• B. Bekka , P. de la Harpe , A. Valette : Kazhdan's Property (T) . Cambridge University Press, 2008. ISBN 978-0-521-88720-5