On the growth of Fourier multipliers (Q6619340)

The paper proves the following theorems.\N\begin{itemize}\N\item[1.] Let \(\Gamma\) be a uniform lattice in \(G=SL(3,\mathbb{R}).\) Then, \(M_0A(\Gamma)\) is a proper subalgebra of \(MA(\Gamma).\)\N\item[2.] Let \(G\) be a uniform lattice in \(G=SL(3,\mathbb{R})\) and let \(\Omega \subseteq G\) be a relatively compact Borel fundamental domain for the right \(\Gamma\)-action of \(G.\) Then, the restriction of the map \[\Phi:\ell^\infty(\Gamma)\rightarrow C_b(G),\ \phi\mapsto 1_G * (\phi_{\mu_\Gamma})*\widetilde{1}_G\] does not define a bounded map \(MA(\Gamma)\rightarrow MA(\Gamma).\)\N\item[3.] Let \(H\) be an unbounded closed subgroup of a locally compact group \(G.\) Suppose that \(H\) satisfies property \((T_{\mbox{schur}},G,H,l)\) for a compact subgroup \(K\) and a proper length function \(l\) of \(G.\) Then, \((H,l|_H)\) does not admit \(K\)-bi-invariant tame cots.\N\item[4.] The following groups have completely bounded characteristic tame sets.\N\begin{itemize}\N\item[(a)] \(\mathbb{Z}^d \rtimes_A \mathbb{Z}\) for any \(d\in\mathbb{N}\) and \(A\in SL(g,\mathbb{Z}).\)\N\item[(b)] \(\mathbb{Z}\left[\frac{1}{pq}\right]\) for any coprime \(p,q\in \mathbb{N}.\)\N\item[(c)] Lamplighter groups \(\mathbb{Z}_p\) wreath product with \(\mathbb{Z}\) for any \(p\in \mathbb{N}.\)\N\item[(d)] Baumslag-Solitar groups \(BS(p,q)=\{a,t|ta^{p}t^{-1}=a^q\}\) for any \(p,q\in \mathbb{N}\)\N\end{itemize}\N\end{itemize}\N\NThe paper is well written.