Property (

Cover -- Title page -- Chapter 1. Introduction -- 1.1. The main result -- 1.2. Property ( ) for \EL_{ }( ): summary of prior work -- 1.3. Almost orthogonality, codistance and a spectral criterion from [12] -- 1.4. Groups graded by root systems and associated graphs of groups -- 1.5. Further examples -- 1.6. Application to expanders -- Concluding remarks -- Chapter 2. Preliminaries -- 2.1. Property ( ) -- 2.2. Relative property ( ) and Kazhdan ratios -- 2.3. Using relative property ( ) -- 2.4. Orthogonality constants, angles and codistances -- 2.5. Kazhdan constants for nilpotent groups and group extensions -- Chapter 3. Generalized spectral criterion -- 3.1. Graphs and Laplacians -- 3.2. Spectral criteria -- 3.3. Proof of Theorem 3.3 -- Chapter 4. Root Systems -- 4.1. Classical root systems -- 4.2. General root systems -- 4.3. Weyl graphs -- 4.4. Groups graded by root systems -- 4.5. A few words about the small Weyl graph -- Chapter 5. Property ( ) for groups graded by root systems -- 5.1. Estimates of codistances in nilpotent groups -- 5.2. Estimates of codistances in Borel subgroups -- 5.3. Norm estimates -- 5.4. Proof of Theorem 5.1 -- Chapter 6. Reductions of root systems -- 6.1. Reductions -- 6.2. Examples of good reductions -- Chapter 7. Steinberg groups over commutative rings -- 7.1. Graded covers -- 7.2. Steinberg groups over commutative rings -- 7.3. Standard sets of generators of Steinberg groups. -- 7.4. Property ( ) for Steinberg groups -- Chapter 8. Twisted Steinberg groups -- 8.1. Constructing twisted groups -- 8.2. Graded automorphisms of _{Φ}^{ }(ℝ) and \St_{Φ}(ℝ) -- 8.3. Unitary Steinberg groups over non-commutative rings with involution -- 8.4. Twisted groups of types ² _{ } and ^{2,2} _{2 -1} -- 8.5. Further twisted examples -- 8.6. Proof of relative property ( ) for type ₂ -- 8.7. Twisted groups of type ² ₄.

8.8. More groups graded by root systems -- Chapter 9. Application: Mother group with property ( ) -- 9.1. Some general reductions -- 9.2. Bounded rank case -- 9.3. Unbounded rank case: overview -- 9.4. Unbounded rank case: proof -- 9.5. Groups of type ₁. -- 9.6. Alternating Groups -- Chapter 10. Estimating relative Kazhdan constants -- 10.1. Hilbert-Schmidt scalar product -- 10.2. Relative property ( ) for group extensions -- 10.3. Codistance bounds in nilpotent groups -- Appendix A. Relative property ( ) for ( _{ }( )⋉ ⁿ, ⁿ) -- Bibliography -- Index -- Back Cover.

The authors introduce and study the class of groups graded by root systems. They prove that if \Phi is an irreducible classical root system of rank \geq 2 and G is a group graded by \Phi, then under certain natural conditions on the grading, the union of the root subgroups is a Kazhdan subset of G. As the main application of this theorem the authors prove that for any reduced irreducible classical root system \Phi of rank \geq 2 and a finitely generated commutative ring R with 1, the Steinberg group {\mathrm St}_{\Phi}(R) and the elementary Chevalley group \mathbb E_{\Phi}(R) have property (T). They also show that there exists a group with property (T) which maps onto all finite simple groups of Lie type and rank \geq 2, thereby providing a "unified" proof of expansion in these groups.

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Electronic reproduction. Ann Arbor, Michigan : ProQuest Ebook Central, 2024. Available via World Wide Web. Access may be limited to ProQuest Ebook Central affiliated libraries.