Let *G* be a discrete group and π a unitary representation of *G* on some Hilbert space $\mathcal{\mathscr{H}}$. For a finite set $F\subset G$ and $\epsilon >0$, a vector $\xi \in \mathcal{\mathscr{H}}$ is $(F,\epsilon )$-invariant if ${\mathrm{max}}_{g\in F}\parallel \pi (g)\xi -\xi \parallel <\epsilon $. Recall that π almost has invariant vectors if, for every pair $(F,\epsilon )$, the group *G* has $(F,\epsilon )$-vectors; and that the group *G* has *Kazhdan’s Property (T)* or is a *Kazhdan group* if every unitary representation of *G* almost having invariant vectors, has non-zero invariant vectors; see e.g. [3] for Property (T).
The definition can be reformulated in terms of weak containment of representations: *G* has Property (T) if every unitary representation weakly containing the trivial representation of *G*, contains it strongly (see [3, Remark 1.1.2]). Crucial for us is an equivalent characterization due to P. S. Wang [12, Corollary 1.9 and Theorem 2.1]: the group *G* has Property (T) if and only if for some (hence every) irreducible finite-dimensional unitary representation σ of *G*, every unitary representation π of *G* that contains σ weakly, contains it strongly.

It is a simple but useful fact that, if *G* has Property (T) and π is a unitary representation almost having invariant vectors, “almost invariant vectors are close to invariant vectors”. More precisely, we have the following.

#### Proposition 1.1 ([3, Proposition 1.1.9]).

*Let **G* be a Kazhdan group. If *S* is a finite generating set of *G* and ${\epsilon}_{\mathrm{0}}$ is the corresponding Kazhdan constant, then for every $\delta \mathrm{\in}\mathrm{]}\mathrm{0}\mathrm{,}\mathrm{1}\mathrm{[}$ and every unitary representation π of *G*, any $\mathrm{(}S\mathrm{,}{\epsilon}_{\mathrm{0}}\mathit{}\delta \mathrm{)}$-invariant vector ξ satisfies $\mathrm{\parallel}\xi \mathrm{-}P\mathit{}\xi \mathrm{\parallel}\mathrm{\le}\delta \mathit{}\mathrm{\parallel}\xi \mathrm{\parallel}$, where *P* is the orthogonal projection on the subspace of $\pi \mathit{}\mathrm{(}G\mathrm{)}$-invariant vectors.

For a Kazhdan group *G* and a unitary representation π of *G*, fix a unit vector ξ and look at the coefficient function

${\varphi}_{\pi ,\xi}(g)=\u3008\pi (g)\xi ,\xi \u3009\mathit{\hspace{1em}}(g\in G).$

The question we first address in this paper is: if ${\varphi}_{\pi ,\xi}$ is close to some coefficient of an irreducible finite-dimensional unitary representation σ of *G*, must ξ be close to a finite-dimensional invariant subspace of π carrying a sub-representation isomorphic to σ? We will see that, in analogy to Proposition 1.1, the answer is positive – with some care.

#### Definition 1.2.

Let *G* be a finitely generated group with a symmetric finite generating set $S\subseteq G$ containing *e* and let ϕ be some normalized positive definite function on *G* associated with a unitary irreducible representation σ, of finite dimension *d*. Let π be some unitary representation of *G* on $\mathcal{\mathscr{H}}$. Let $\epsilon >0$. Say that a unit vector $\xi \in \mathcal{\mathscr{H}}$ is $(\pi ,\varphi ,\epsilon )$-good if for every $s\in {S}^{2{d}^{2}+1}$ we have $|{\varphi}_{\pi ,\xi}(s)-\varphi (s)|<\epsilon $.

Note that ${S}^{k}$ is just the ball of radius *k* centered at the identity in *G*. So there is a certain lack of uniformity in Definition 1.2: we require an approximation of ${\varphi}_{\pi ,\xi}$ by ϕ on a ball whose size depends on the dimension of the representation *d*. Our main result, proved in Section 2, can be viewed as a quantitative version of Wang’s result.

#### Theorem 1.3.

*Let **G* be a discrete Kazhdan group, *S* a finite symmetric generating set with $e\mathrm{\in}S$, and let ϕ be a normalized positive definite function on *G* associated with a finite-dimensional unitary irreducible representation σ of *G*. For every $\delta \mathrm{\in}\mathrm{]}\mathrm{0}\mathrm{,}\mathrm{1}\mathrm{[}$ there exists ${\epsilon}_{\varphi \mathrm{,}\delta}\mathrm{>}\mathrm{0}$ such that for every unitary representation π of *G* on a Hilbert space $\mathcal{H}$, and every unit vector $x\mathrm{\in}\mathcal{H}$ that is $\mathrm{(}\pi \mathrm{,}\varphi \mathrm{,}{\epsilon}_{\varphi \mathrm{,}\delta}\mathrm{)}$-good, there exists a unit vector ${x}^{\mathrm{\prime}}\mathrm{\in}\mathcal{H}$ with $\mathrm{\parallel}x\mathrm{-}{x}^{\mathrm{\prime}}\mathrm{\parallel}\mathrm{\le}\delta $ such that the restriction of π to the span of $\pi \mathit{}\mathrm{(}G\mathrm{)}\mathit{}{x}^{\mathrm{\prime}}$ is isomorphic to σ.

In Section 3, we apply Theorem 1.3 to the study of the global structure of the space of unitary representations of Kazhdan groups. Let us start with the notation. Let *G* be an arbitrary countable group and let $\mathcal{\mathscr{H}}$ be a separable infinite-dimensional Hilbert space. The set $\mathrm{Rep}(G,\mathcal{\mathscr{H}})$ of all homomorphisms from *G* into the unitary group $U(\mathcal{\mathscr{H}})$ can be viewed as a closed subset of the product space $U{(\mathcal{\mathscr{H}})}^{G}$, when we equip $U(\mathcal{\mathscr{H}})$ with the strong operator topology. With this identification, $\mathrm{Rep}(G,\mathcal{\mathscr{H}})$ is a Polish (i.e. separable and completely metrizable) space. We refer the reader to the work [8], especially to the section on the spaces of unitary representations, for more information about this point of view on unitary representations. Recall that two unitary representations ${\pi}_{1},{\pi}_{2}\in \mathrm{Rep}(G,\mathcal{\mathscr{H}})$ are *isomorphic*, or *unitarily equivalent* if there is a unitary operator $\varphi \in U(\mathcal{\mathscr{H}})$ such that ${\pi}_{1}(g)=\varphi {\pi}_{2}(g){\varphi}^{*}$ for every $g\in G$. Notice that this is an orbit equivalence relation given by the action of the unitary group $U(\mathcal{\mathscr{H}})$ on the space $\mathrm{Rep}(G,\mathcal{\mathscr{H}})$ by conjugation. Kechris raised the question (see again the section on the space of unitary representations in [8]) of whether or not there are countable groups with a *generic unitary representation*, where “generic” here means its conjugacy class is large in the sense of Baire category, i.e. a representation whose class under the unitary equivalence contains a dense ${G}_{\delta}$ subset. As a matter of fact, we note that it follows from the topological zero-one law that for every countable group *G* either there is a generic representation in $\mathrm{Rep}(G,\mathcal{\mathscr{H}})$, or all conjugacy classes are meager (see e.g. [7, Theorem 8.46]; to apply it, note that there is a dense conjugacy class in $\mathrm{Rep}(G,\mathcal{\mathscr{H}})$ – indeed, take some countable dense set of representations from $\mathrm{Rep}(G,\mathcal{\mathscr{H}})$ and consider their direct sum).

Here as an application of Theorem 1.3 we prove the following result.

#### Theorem 1.4.

*Let **G* be a discrete Kazhdan group such that finite-dimensional representations are dense in the unitary dual $\widehat{G}$. Then there is a generic unitary representation of *G*.

We note that, although not explicitly stated there, this result already follows from a more general result of Kerr, Li and Pichot from [9], where they prove (see Theorem 2.5 there) that if *A* is a separable ${C}^{*}$-algebra where finite-dimensional representations are dense in $\widehat{A}$, then there is a dense ${G}_{\delta}$ class in $\mathrm{Rep}(A,\mathcal{\mathscr{H}})$. Theorem 1.4 is then a special case for $A={C}^{*}(G)$. Our proof is nevertheless done by more elementary means, in particular it does not invoke Voiculescu’s theorem (see [9, proof of Theorem 2.5] for details).

The converse of Theorem 1.4 can be also derived from [9], see Remark 3.1 below.

Another open question posed by Kechris in [8, Problem H.16] is whether the subset of those representations $\pi \in \mathrm{Rep}(G,\mathcal{\mathscr{H}})$, where *G* is still a countable group, that are equivalent to Koopman representations, is meager in $\mathrm{Rep}(G,\mathcal{\mathscr{H}})$ or not. Such representations are called *realizable by an action* in [8]. Let us recall the terminology first. Let $(X,\mu )$ be a standard probability space (i.e. a space isomorphic to the unit interval $[0,1]$ equipped with the Lebesgue measure). Let $\alpha :G\curvearrowright (X,\mu )$ be an action of a countable group *G* on *X* by measure-preserving measurable transformations. Consider the unitary representation ${\pi}_{\alpha}:G\to {L}^{2}(X,\mu )$ defined by ${\pi}_{\alpha}(g)f(x)=f(\alpha ({g}^{-1},x))$ for every $f\in {L}^{2}(X,\mu )$. The *Koopman representation* of α is the restriction of ${\pi}_{\alpha}$ to the invariant subspace ${L}_{0}^{2}(X,\mu )$, which is the orthogonal complement of the invariant subspace of constant functions.

In Section 4 we prove the following result addressing the question of Kechris.

#### Theorem 1.5.

*Let **G* be a discrete Kazhdan group such that finite-dimensional representations are dense in the unitary dual $\widehat{G}$. Then the set of representations realizable by an action is comeager in $\mathrm{Rep}\mathit{}\mathrm{(}G\mathrm{,}\mathcal{H}\mathrm{)}$.

Let us record that the condition that finite-dimensional representations are dense in the unitary dual $\widehat{G}$ is, by the result of Archbold from [1], equivalent with the statement that the full group ${C}^{*}$-algebra ${C}^{*}(G)$ is residually finite-dimensional. That is in turn, by the result of Exel and Loring from [6] (see also [11]), equivalent with the statement that finite-dimensional representations are dense in $\mathrm{Rep}(G,\mathcal{\mathscr{H}})$, which we shall use in the proof. Note that we call a representation $\pi \in \mathrm{Rep}(G,\mathcal{\mathscr{H}})$ finite-dimensional if the subalgebra $\pi (G)$ generates in $B(\mathcal{\mathscr{H}})$ is finite-dimensional.

The existence of infinite discrete Kazhdan groups with residually finite-dimensional ${C}^{*}$-algebras seems to be open – see [3, Question 7.10] and also Lubotzky and Shalom [10, Question 6.5] where they ask if there are infinite discrete Kazhdan groups with property FD, which is strictly stronger than having a residually finite-dimensional ${C}^{*}$-algebra (a group has *property FD* if representations factoring through finite groups are dense in the unitary dual).

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