Kazhdan's property (T) and related properties of locally compact and discrete groups

• Main Author: Deutsch, Annabel
• Published: University of Edinburgh 1992
• Subjects: 510
• Online Access: http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.649494

In this thesis we look at a number of properties related to Kazhdan's property (T), for a locally compact, metrisable, σ-compact group. For such a group, <i>G</i>, the following properties are equivalent. 1. Kazhdan's definition of property (T): the trivial representations is isolated in the unitary dual of <i>G</i> (with the Fell topology). 2. The group, <i>G</i>, is compactly generated and for every compact generating set, <i>K</i>, there is a positive constant, ε, such that if π is a unitary representation of <i>G</i> on a Hilbert space, cal H, and ζ is a unit vector in cal H such that vskip 0.7cmthen π fixes some non-zero vector in cal H. This is often taken as the definition of property (T). 3. Every conditionally negative type function on <i>G</i> is bounded. 4. For a discrete group, <i>G</i> is finitely generated and for every finite generating set, <i>K</i>, zero is an isolated point in the spectrum of the Laplacian. From 2 we can define the Kazhdan constant, the largest possible value of ε for a given <i>G</i> and <i>K</i>. In Chapter 2 we investigate how to calculate these constants. In Chapter 4 we look at the bound on conditionally negative type functions and use its existence to extend a result of A.Connes and V.Jones about the von Neumann algebras of property (T) groups. The first half of Chapter 3 examines the spectrum of the Laplacian for a discrete group and finite generating set and compares its least positive element to the Kazhdan constant. Non-compact property (T) groups are all non-amenable. However, the standard example of a non-amenable group <i>F</i>_2, does not have property (T). The second half of Chapter 3 looks at the spectrum of λ(Δ) for F_2 with various generating sets, where λ is the left regular representation of <i>G</i> on <i>l</i><SUP>2</SUP>(<i>G</i>). For any non-amenable group, the smallest element of Spλ(Δ) is positive. Chapter 5 is an attempt to extend various results about <i>F</i><SUB>2</SUB> to other non-amenable groups.