On relative property (T) (Q2908717)

The relative property (T) is defined for a pair \((G,L)\) where \(G\) is a locally compact group and \(L\) is its subgroup. \((G,L)\) has this property if every continuous unitary representation of \(G\) which has almost invariant vectors has also \(L\)-invariant vectors. Being weaker than Kazhdan's property (T), it has various applications, notably in the rigidity theory of group actions (constructions of S.~Popa).NEWLINENEWLINEThe aim of this paper is to find new examples of group pairs with relative property (T). It can be considered as a development of the results of \textit{T.~Fernós} [Ann. Inst. Fourier 56, No. 6, 1767--1804 (2006; Zbl 1175.22004)].NEWLINENEWLINEThe main results are as follows. Let \(S\) be a finite set of primes, and \(\mathbb Z[S]\) the ring obtained by inverting the primes in \(S\). Corollary 1.5 states that, for every natural number \(N\) and every nonempty finite \(S\), there is a finitely generated free group \(\Gamma< SL_N(\mathbb Z[S])\) and a morphism \(\rho: \Gamma\to GL_N(\mathbb Z[S])\) such that the pair \((\Gamma \ltimes \mathbb Z[S]^N, \mathbb Z[S]^N)\) has relative property (T).NEWLINENEWLINEThe second result to cite, Corollary 1.7, states that if \(G\) is a connected real algebraic semisimple Lie group without compact factors and \(\Gamma<G\) is an arithmetic lattice, then there is a linear representation \(\Gamma'\to SL_N(\mathbb Z)\) of a finite-index subgroup \(\Gamma'<\Gamma\) such that \((\Gamma'\ltimes\mathbb Z^N,\mathbb Z^N)\) has relative property (T).NEWLINENEWLINEFor the entire collection see [Zbl 1225.00042].