Title:

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

In this thesis we look at a number of properties related to Kazhdan's property (T), for a locally compact, metrisable, σcompact group. For such a group, G, the following properties are equivalent. 1. Kazhdan's definition of property (T): the trivial representations is isolated in the unitary dual of G (with the Fell topology). 2. The group, G, is compactly generated and for every compact generating set, K, there is a positive constant, ε, such that if π is a unitary representation of G on a Hilbert space, cal H, and ζ is a unit vector in cal H such that vskip 0.7cmthen π fixes some nonzero vector in cal H. This is often taken as the definition of property (T). 3. Every conditionally negative type function on G is bounded. 4. For a discrete group, G is finitely generated and for every finite generating set, K, 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 G and K. 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. Noncompact property (T) groups are all nonamenable. However, the standard example of a nonamenable group F_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 G on l^{2}(G). For any nonamenable group, the smallest element of Spλ(Δ) is positive. Chapter 5 is an attempt to extend various results about F_{2} to other nonamenable groups.
