Kazhdan pairs, Kazhdan sets and Kazhdan constants

The definitions of Kazhdan pair and Kazhdan set apply to every subset $Q\subseteq G$, including $Q=\mathrm{\varnothing }$. When $Q=\mathrm{\varnothing }$, every unit vector is $(Q,\epsilon )$-invariant, so the pair condition requires every representation with a unit vector to have a nonzero invariant vector.

Kazhdan-pair parameters are downward closed: if $(Q,\delta )$ is a pair and $0<\epsilon \le \delta$, then each $(Q,\epsilon )$-invariant unit vector is also $(Q,\delta )$-invariant, so $(Q,\epsilon )$ is a pair. Thus, with the zero adjoined in the definition, $\kappa (G,Q)$ is a well-defined extended-real threshold, equals $0$ if there is no positive Kazhdan parameter, and equals $+\mathrm{\infty }$ if every positive parameter is Kazhdan.

For a unit vector $\xi$, each displacement is at most $2$ by [F2]. If $Q\mathrm{\ne }\mathrm{\varnothing }$, its displacement values form a nonempty subset of $[0,2]$ and have a real supremum by [F4]; if $Q=\mathrm{\varnothing }$, $d_Q(\pi ,\xi )=0$ by definition. Thus $\kappa (G,Q,\pi )\in [0,2]$ for $H\mathrm{\ne }\{0\}$. If $H=\{0\}$, there are no unit vectors, so the declared extended-real empty-infimum convention gives $\kappa (G,Q,\pi )=+\mathrm{\infty }$.

If $Q$ is compact and nonempty, for every unit vector $\xi$ the function $x\mapsto \mathrm{\parallel }\pi (x)\xi -\xi \mathrm{\parallel }$ is continuous: the orbit map is continuous by [F1], and the norm is continuous by the reverse triangle inequality in [F2]. Its image on $Q$ is compact and therefore has a maximum by [F3]. For $Q=\mathrm{\varnothing }$, the explicitly assigned displacement $0$ serves as its maximum convention.

By the definition of extended-real supremum, every pair parameter $\epsilon$ satisfies $\epsilon \le \kappa (G,Q)$, and if $0<\epsilon <\kappa (G,Q)$ then some pair parameter $\delta$ has $\delta >\epsilon$. Downward closure from step 1.2 makes $(Q,\epsilon )$ a pair.

For compact $Q$, a unit vector is $(Q,\epsilon )$-invariant exactly when $d_Q(\pi ,\xi )<\epsilon$: in the nonempty case this follows because the continuous displacement attains its maximum, and in the empty case both conditions hold for every $\epsilon >0$. Consequently, for a representation on a nonzero Hilbert space with no invariant vector, it has no such unit vector exactly when $\kappa (G,Q,\pi )\ge \epsilon$.

Suppose $\pi$ has no nonzero invariant vector and $(Q,\epsilon )$ is a Kazhdan pair. For every unit vector $\xi$, $d_Q(\pi ,\xi )\ge \epsilon$: otherwise every $x\in Q$ would have displacement $<\epsilon$, making $\xi$ a $(Q,\epsilon )$-invariant unit vector and forcing a nonzero invariant vector. This includes $Q=\mathrm{\varnothing }$, where such a representation cannot exist on a nonzero Hilbert space. Taking the infimum over unit vectors, then the supremum over pair parameters, gives $\kappa (G,Q,\pi )\ge \kappa (G,Q)$.

The zero-space representation has no unit vectors and value $+\mathrm{\infty }$, so it never obstructs a Kazhdan pair. For compact $Q$, step 2.2 therefore says that $(Q,\epsilon )$ is a pair exactly when every representation without nonzero invariant vectors has displacement constant at least $\epsilon$, equivalently when their extended-real infimum is at least $\epsilon$. Taking the supremum of $\{0\}$ together with the admissible pair parameters yields $\kappa (G,Q)={\inf }_{\pi }\kappa (G,Q,\pi )$.

Steps 2.1 and 3.1 establish the threshold statements for arbitrary $Q$; steps 1.4–3.2 establish the endpoint equivalence and the infimum formula for compact $Q$, with empty $Q$ and the zero-space representation handled by the stated conventions.