Normal relative property (T) controls the distance to the invariant subspace

Suppose, toward a contradiction, that relative property (T) holds but no relative Kazhdan pair has compact first component. The compact subsets of $G$ form a set $\mathcal{K}(G)\subseteq \mathcal{P}(G)$, and ${\mathbb{N}}_{>0}$ is a set. For each $(K,m)\in \mathcal{K}(G)\times {\mathbb{N}}_{>0}$, let ${\mathcal{F}}_{K,m}\subseteq {\mathbb{C}}^G$ consist of the normalized continuous positive-type coefficient functions of strongly continuous representations with no nonzero $N$-invariant vector and a unit vector that is $(K,1\mathrm{/}m)$-invariant. Failure of every compact relative Kazhdan pair makes each ${\mathcal{F}}_{K,m}$ nonempty. By AC choose ${\phi }_{K,m}\in {\mathcal{F}}_{K,m}$ for all $(K,m)$. For any witness $({\pi }^w,H^w,{\xi }^w)$ realizing ${\phi }_{K,m}$, equality of the coefficients gives equality of the Gram matrices on finite orbit sums; thus ${\sum }_g{c}_g{\pi }_{{\phi }_{K,m}}(g){\eta }_{{\phi }_{K,m}}\mapsto {\sum }_g{c}_g{\pi }^w(g){\xi }^w$ is a well-defined isometry of cyclic spans, extends to a unitary onto the witness's cyclic carrier, and intertwines the representations. The carrier has no nonzero $N$-invariant vector, while its cyclic unit vector is $(K,1\mathrm{/}m)$-invariant; hence the canonical GNS representation has these same properties. The Hilbert direct sum over this set-indexed family has no nonzero $N$-invariant vector, while for every compact $K$ and every $\eta >0$ a coordinate with $1\mathrm{/}m<\eta$ supplies a $(K,\eta )$-invariant unit vector. Thus the direct sum has almost invariant vectors, contradicting relative property (T). Therefore some compact $K_0$ and ${\epsilon }_0>0$ form a relative Kazhdan pair.

The fixed-vector subspace is $H^N={\bigcap }_{n\in N}\ker (\pi (n)-I)$. Every kernel is closed because $\pi (n)-I$ is continuous linear, so $H^N$ is a closed linear subspace. It is $G$-invariant: if $v\in H^N$, then for $g\in G$ and $n\in N$, $\pi (n)\pi (g)v=\pi (g)\pi (g^{-1}ng)v=\pi (g)v$ because $N$ is normal. Its orthogonal complement is also $G$-invariant: if $w\perp H^N$ and $v\in H^N$, then $⟨\pi (g)w,v⟩=⟨w,\pi (g^{-1})v⟩=0$.

By AC and [F4], write $\xi ={\xi }_N+{\xi }_{\perp }$ with ${\xi }_N\in H^N$ and ${\xi }_{\perp }\in (H^N{)}^{\perp }$. Pythagoras shows that $\mathrm{dist}(\xi ,H^N)=\mathrm{\parallel }{\xi }_{\perp }\mathrm{\parallel }$: for every $v\in H^N$, $\mathrm{\parallel }\xi -v{\mathrm{\parallel }}^2=\mathrm{\parallel }{\xi }_N-v{\mathrm{\parallel }}^2+\mathrm{\parallel }{\xi }_{\perp }{\mathrm{\parallel }}^2$, with equality at $v={\xi }_N$.

The restriction to $(H^N{)}^{\perp }$ is strongly continuous and has no nonzero $N$-invariant vector, since such a vector would lie in both $H^N$ and $(H^N{)}^{\perp }$. If $Q=\mathrm{\varnothing }$, any unit vector in this restriction is vacuously $(Q,\epsilon )$-invariant, so [F1] forces $(H^N{)}^{\perp }=\{0\}$. If $Q\mathrm{\ne }\mathrm{\varnothing }$, no unit vector in the restriction can be $(Q,\epsilon )$-invariant by [F1]; hence every nonzero $w\in (H^N{)}^{\perp }$ has some $q\in Q$ with $\mathrm{\parallel }\pi (q)w-w\mathrm{\parallel }\ge \epsilon \mathrm{\parallel }w\mathrm{\parallel }$.

For the final clause, choose the compact relative pair $(K_0,{\epsilon }_0)$ from step 1.1. If $K_0=\mathrm{\varnothing }$, [F1] forces every representation without nonzero $N$-invariants to be the zero representation, and any $\delta >0$ works. Otherwise suppose no uniform $\delta$ exists. For every positive integer $m$ there is a representation without nonzero $N$-invariants and a unit vector with ${\mathrm{\Delta }}_Q<1\mathrm{/}m$, by normalizing a nonzero vector violating the proposed bound $1\mathrm{/}m$. As in step 1.1, AC chooses their normalized coefficient functions ${\phi }_m$ from nonempty subsets of ${\mathbb{C}}^G$, and canonical GNS gives representations $({\pi }_m,H_m)$ with unit vectors ${\xi }_m$, no nonzero $N$-invariants and ${\mathrm{\Delta }}_Q^{{\pi }_m}({\xi }_m)<1\mathrm{/}m$. The Gram-isometry argument of step 1.1 transfers all these properties from any witness. For each $g\in Q^k$, telescoping gives $\mathrm{\parallel }{\pi }_m(g){\xi }_m-{\xi }_m\mathrm{\parallel }\le k{\mathrm{\Delta }}_Q^{{\pi }_m}({\xi }_m)<k\mathrm{/}m$; therefore [F14] gives $\mathrm{\mid }{\phi }_m(g)-1\mathrm{\mid }\le \mathrm{\parallel }{\pi }_m(g){\xi }_m-{\xi }_m\mathrm{\parallel }\to 0$. Since the positive powers of $Q$ cover $G$, ${\phi }_m(g)\to 1$ at every $g\in G$.

Since both orthogonal summands are $G$-invariant, Pythagoras gives, for every $q\in Q$, $\mathrm{\parallel }\pi (q)\xi -\xi {\mathrm{\parallel }}^2=\mathrm{\parallel }\pi (q){\xi }_N-{\xi }_N{\mathrm{\parallel }}^2+\mathrm{\parallel }\pi (q){\xi }_{\perp }-{\xi }_{\perp }{\mathrm{\parallel }}^2$. For nonempty $Q$ and ${\xi }_{\perp }\mathrm{\ne }0$, step 2.2 supplies a $q$ whose second term is at least ${\epsilon }^2\mathrm{\parallel }{\xi }_{\perp }{\mathrm{\parallel }}^2$, hence ${\mathrm{\Delta }}_Q^{\pi }(\xi )\ge \epsilon \mathrm{\parallel }{\xi }_{\perp }\mathrm{\parallel }$. If $Q$ is empty or ${\xi }_{\perp }=0$, the same inequality follows from step 2.2. By step 2.1 this proves the first bound.

Define $T:K_0\to {\mathbb{C}}^{{\mathbb{N}}_{>0}}$ by $T(g)=({\phi }_m(g){)}_m$ and put $X=T(K_0)$ with the product subspace topology. By [F10], $X$ is compact. It is Hausdorff: distinct points differ in some coordinate, whose distinct complex values have disjoint open disks; the inverse images of those disks separate the points. Thus $X$ is LCH, because the whole compact space is a neighborhood of each point. Its coordinate functions $f_m(x)=x_m$ are continuous, satisfy $\mathrm{\mid }{f}_m\mathrm{\mid }\le 1$ by [F14], and converge pointwise to $1$ by step 2.3. Every continuous function on $X$ is bounded by compactness and has compact positive superlevel sets (closed subsets of $X$), so $C(X;\mathbb{C})=C_0(X;\mathbb{C})$ with its supremum norm. For any bounded complex-linear functional $L$, [F11] supplies a finite regular complex measure $\mu$ representing it. Applying [F12] with the finite measure $\mathrm{\mid }\mu \mathrm{\mid }$ and the constant majorant $2$ gives ${\int }_X\mathrm{\mid }{f}_m-1\mathrm{\mid }d\mathrm{\mid }\mu \mathrm{\mid }\to 0$, hence $\mathrm{\mid }L(f_m)-L(1)\mathrm{\mid }\le {\int }_X\mathrm{\mid }{f}_m-1\mathrm{\mid }d\mathrm{\mid }\mu \mathrm{\mid }\to 0$. Convergence for each functional implies convergence on every finite list of functionals, so $f_m\to 1$ weakly in $C(X;\mathbb{C})$.

For $n\in N$, $\pi (n){\xi }_N={\xi }_N$, so $\mathrm{\parallel }\pi (n)\xi -\xi \mathrm{\parallel }=\mathrm{\parallel }\pi (n){\xi }_{\perp }-{\xi }_{\perp }\mathrm{\parallel }\le 2\mathrm{\parallel }{\xi }_{\perp }\mathrm{\parallel }$ by [F2, F5]. Taking the supremum over $n$ and applying steps 2.1 and 3.1 proves the second inequality, including $N=\{e\}$ and ${\xi }_{\perp }=0$.

The constant $1$ lies in the weak closure of the convex hull of $\{f_m:m\ge 1\}$ and therefore, by [F13], in its norm closure. Choose a finite convex combination $f={\sum }_{j=1}^r{a}_j{f}_{m_j}$ with $a_j\ge 0$, ${\sum }_j{a}_j=1$ and $\mathrm{\parallel }f-1{\mathrm{\parallel }}_{\mathrm{\infty }}<{\epsilon }_0^2\mathrm{/}2$. The finite direct sum $\rho ={⨁}_{j=1}^r{\pi }_{m_j}$ is strongly continuous, has no nonzero $N$-invariant vector, and has the unit vector $\eta =(\sqrt{a_j}{\xi }_{m_j}{)}_j$. Its coefficient is ${\sum }_j{a}_j{\phi }_{m_j}(g)=f(T(g))$ for $g\in K_0$. Thus $\mathrm{\parallel }\rho (g)\eta -\eta {\mathrm{\parallel }}^2=2(1-\mathrm{Re}f(T(g)))<{\epsilon }_0^2$ for every $g\in K_0$, contradicting the relative pair. A uniform $\delta >0$ consequently exists for unit vectors; homogeneity gives the asserted inequality for every nonzero vector, while the zero vector is immediate. The positive-power hypothesis excludes $Q=\mathrm{\varnothing }$ since $G$ contains its identity. No bounded-word-length claim on $K_0$, Hausdorff hypothesis on $G$, or identity-neighborhood hypothesis on $Q$ was used.