Relative property (T) for SL2(R) semidirect R2

The matrix group $K$ is the determinant-one closed subset of ${\mathbb{R}}^4$, so it is locally compact Hausdorff by [F1] and second countable by hereditary second countability. The additive group $V={\mathbb{R}}^2$ has the same topological properties, and the finite product $K\times V$ is second countable.

Suppose no ${\epsilon }_0\in (0,1]$ works. For each $m\ge 1$, a strongly continuous representation without a nonzero $N$-invariant vector and a unit vector with $Q_0$-displacement less than $1\mathrm{/}m$ then exist. Let ${\mathcal{F}}_m\subseteq {\mathbb{C}}^G$ be the set of all diagonal coefficient functions of such witnesses. Each ${\mathcal{F}}_m$ is nonempty and is a set; by AC choose ${\phi }_m\in {\mathcal{F}}_m$.

Under the identification in [F4], the dual action is $k\cdot \xi =(k^{-1}{)}^T\xi$. Thus the matrices acting on the dual for $u_{+}$ and $u_{-}$ are respectively $a_{+}=(u_{+}^{-1}{)}^T=\left(\begin{array}{cc}1& 0\\ -1& 1\end{array}\right)$ and $a_{-}=(u_{-}^{-1}{)}^T=\left(\begin{array}{cc}1& -1\\ 0& 1\end{array}\right)$.

Under the coordinate swap in [F2], the stated multiplication is the external semidirect-product law. The action $(k,v)\mapsto kv$ is continuous because its coordinates are finite sums of products of matrix and vector coordinates; multiplication $(k,v)(k^{\mathrm{\prime }},v^{\mathrm{\prime }})=(k{k}^{\mathrm{\prime }},v+k{v}^{\mathrm{\prime }})$ and inversion $(k,v{)}^{-1}=(k^{-1},-k^{-1}v)$ are continuous. Conjugation gives $(k,v)(I,w)(k,v{)}^{-1}=(I,kw)$, so $N$ is normal; it is closed as $\{I\}\times V$ in the Hausdorff product. Thus $G$ is a topological group.

Let $({\pi }_m,H_m,{\eta }_m)$ be the GNS triple of ${\phi }_m$. For any witness $({\pi }_m^w,H_m^w,{\xi }_m^w)$ realizing the coefficient ${\phi }_m$, the map ${\sum }_g{c}_g{\pi }_m(g){\eta }_m\mapsto {\sum }_g{c}_g{\pi }_m^w(g){\xi }_m^w$ preserves inner products because both cyclic-vector coefficients equal ${\phi }_m$; it is therefore a well-defined isometry of cyclic spans, extends to a unitary onto the witness's cyclic carrier, and intertwines the representations. That carrier has no nonzero $N$-invariant vector, so neither does the GNS representation; ${\eta }_m$ is a unit vector with $Q_0$-displacement less than $1\mathrm{/}m$. Since its cyclic orbit map is continuous and $G$ is second countable, it has a countable dense subset by [F1], and rational complex linear combinations show $H_m$ is separable.

Set ${\rho }_m(v)={\pi }_m(I,v)$ for $v\in V$ and ${\tau }_m(k)={\pi }_m(k,0)$ for $k\in K$. The semidirect law gives ${\tau }_m(k){\rho }_m(v){\tau }_m(k{)}^{-1}={\rho }_m(kv)$. By [F1]–[F3], the spectral-measure lemma applies to $N=V$, $K={\mathrm{S}\mathrm{L}}_2(\mathbb{R})$ and $H_m$: it gives a regular PVM $E_m$ on $\widehat{V}$ such that ${\rho }_m(v)=\int \chi (v)d{E}_m(\chi )$ and ${\tau }_m(k)E_m(B){\tau }_m(k{)}^{-1}=E_m(k\cdot B)$.

For a vector $v$ invariant under all of $N$, let ${\mu }_v(B)=⟨E_m(B)v,v⟩$. For every $q\in {\mathbb{Q}}^2$, the integrated PVM formula and [F5] give $0=\mathrm{\parallel }{\rho }_m(q)v-v{\mathrm{\parallel }}^2={\int }_{\widehat{V}}\mathrm{\mid }\chi (q)-1{\mathrm{\mid }}^{2}d{\mu }_v(\chi )$. For each positive integer $r$, the Borel set $B_{q,r}=\{\chi :\mathrm{\mid }\chi (q)-1\mathrm{\mid }\ge 1\mathrm{/}r\}$ has measure zero, since the integrand is at least $r^{-2}$ there. Their countable union is $\{\chi :\chi (q)\mathrm{\ne }1\}$, hence this set is null. The set ${\mathbb{Q}}^2$ is countable and dense by [F1]; intersecting the corresponding full-measure sets shows that ${\mu }_v$ is concentrated on characters trivial on ${\mathbb{Q}}^2$. Continuity of characters makes such a character trivial on ${\mathbb{R}}^2$, so under [F4] it is the point $0$. Consequently ${\mu }_v(\widehat{V}\setminus \{0\})=0$ and $\mathrm{\parallel }{E}_m(\widehat{V}\setminus \{0\})v{\mathrm{\parallel }}^2={\mu }_v(\widehat{V}\setminus \{0\})=0$, so the projection identity and $E_m(\widehat{V})=I$ imply $v=E_m(\{0\})v$. Conversely, the integrated formula shows every vector in $\mathrm{ran}{E}_m(\{0\})$ is $N$-invariant. Thus $\mathrm{ran}{E}_m(\{0\})=H_m^N$, which is zero by the choice of ${\pi }_m$.

Since ${\eta }_m$ is a unit vector, ${\mu }_m(B):=⟨E_m(B){\eta }_m,{\eta }_m⟩$ is a Borel probability; step 5.1 gives ${\mu }_m(\{0\})=0$. By [F4], identify $\widehat{V}\setminus \{0\}$ homeomorphically with ${\mathbb{R}}^2\setminus \{0\}$, then push forward under the continuous projective map $\mathrm{\Phi }(y)=[y]$ to obtain a Borel probability ${\nu }_m$ on the compact metric space $P^1(\mathbb{R})$.

Let $k\in \{u_{+},u_{-}\}$ and let $B\subseteq \widehat{V}$ be Borel. Put $P=E_m(B)$ and $\zeta ={\tau }_m(k){\eta }_m$. Since $P$ is a contractive projection, expansion in the first inner-product variable and Cauchy–Schwarz give $\mathrm{\mid }{\mu }_{\zeta }(B)-{\mu }_{{\eta }_m}(B)\mathrm{\mid }=\mathrm{\mid }⟨P(\zeta -{\eta }_m),\zeta ⟩+⟨P{\eta }_m,\zeta -{\eta }_m⟩\mathrm{\mid }\le 2\mathrm{\parallel }\zeta -{\eta }_m\mathrm{\parallel }<2\mathrm{/}m$. Covariance gives ${\mu }_{\zeta }(B)={\mu }_{{\eta }_m}(k^{-1}\cdot B)$; because $\mathrm{\Phi }$ intertwines the dual action with the projective action of $k\cdot \xi =(k^{-1}{)}^T\xi$, it follows for every Borel $A\subseteq P^1(\mathbb{R})$ that $\mathrm{\mid }{\nu }_m(k^{-1}\cdot A)-{\nu }_m(A)\mathrm{\mid }<2\mathrm{/}m$.

By [F7], pass to a subsequence ${\nu }_{m_j}$ converging weakly to a Borel probability $\nu$. Fix a continuous real $f$ and an approximation tolerance $\delta >0$; partition its bounded range into finitely many intervals of length at most $\delta$ to obtain a finite-valued Borel simple function $s={\sum }_{l=1}^L{c}_l{\mathbf{1}}_{A_l}$ with $\mathrm{\parallel }f-s{\mathrm{\parallel }}_{\mathrm{\infty }}\le \delta$. For either projective map $a_{+}$ or $a_{-}$, step 7.1 bounds the difference of the integrals of $s$ against the pushed-forward and original ${\nu }_{m_j}$ by $(2\mathrm{/}{m}_j){\sum }_l\mathrm{\mid }{c}_l\mathrm{\mid }$, while [F8] bounds the two approximation errors by $2\delta$ in total. Taking $j\to \mathrm{\infty }$, using weak convergence and continuity of the projective maps, gives $\mathrm{\mid }\int fd(a_{\ast }\nu )-\int fd\nu \mathrm{\mid }\le 2\delta$; as $\delta$ is arbitrary, the integrals are equal. Both measures are regular by [F7], so the uniqueness theorem there implies $(a_{+}{)}_{\ast }\nu =\nu =(a_{-}{)}_{\ast }\nu$.

The matrices $a_{+}$ and $a_{-}$ in step 1.3 are nonidentity unipotents because each difference from $I$ is nonzero and square-zero; their fixed lines are respectively $\mathbb{R}(0,1)$ and $\mathbb{R}(1,0)$, which are distinct. Step 8.1 therefore contradicts [F6].

The contradiction shows that some integer $m\ge 1$ has no counterexample, so ${\epsilon }_0=1\mathrm{/}m\in (0,1]$ makes $(Q_0,{\epsilon }_0)$ a relative Kazhdan pair for $(G,N)$. By [F11], $Q_0$ is compact; hence almost invariant vectors supply a $(Q_0,{\epsilon }_0)$-invariant unit vector, and the pair conclusion gives relative property (T).