A well-known application of Kazhdan's Property (T) is the construction of expander graphs. Background on this is discussed, for example, in this post on Terry Tao's blog. Essentially, Cayley graphs of finite quotients of property (T) groups can give us families of expanders (see Exercise 14 of Tao's blog post). The construction seems to critically use the finite quotients to obtain the unitary representations required to employ the definition of property (T). It would be very nice to have an answer to the following:
Question: Is any of the behavior of expander graphs reflected in the (infinite) Cayley graph of a property (T) group with respect to a finite, symmetric generating set?
Please provide a reference to anything in the literature that sheds some light on this.
A somewhat broader (related) question that may be helpful:
What are some qualitative properties of the Cayley graph of a property (T) group?
For example, does the Cayley graph of a property (T) group exhibit any sort of (local) concentration of measure phenomena using the word metric w.r.t. a finite generating set?
What are some useful intuitions for the Cayley graph of a property (T) group? (Here I'm wondering if there is anything akin to the image of "thin triangles" for hyperbolic groups.)
Best Answer
If Kazhdan's property (T) is reflected in the structure of the Cayley graph, then not in a very geometric way.
Steve Gersten (that is what I read in the book by B. Bekka, P. de la Harpe and A. Valette) was the first who found that Kazhdan's property (T) is not invariant under quasi-isometry. The reason is not complicated. If a central extension $$1 \to \mathbb Z \to \Gamma \to \Lambda $$ is obtained from a bounded cocycle $c \colon \Lambda \times \Lambda \to \mathbb Z$, then $\Gamma$ is quasi-isometric to $\Lambda \times \mathbb Z$. This situation arises for $\Lambda$ a cocompact lattice in a simple real Lie group with infinite fundamental group; such as $SU(2,2)$. Then, $\Gamma$ is the inverse image of $\Lambda$ in the universal covering. In this situation, one actually obtains (for suitable generating sets) a bi-Lipschitz equivalence of Cayley graphs.
Now, $\Lambda \times \mathbb Z$ does not have Kazhdan's property (T) since it surjects onto $\mathbb Z$, but $\Gamma$ has Kazhdan's property (T) inheriting it from the universal cover of $SU(2,2)$.