Quasi-morphisms via local action data and quasi-isometries (Q2796969)

Given a group \(G\), a \textit{quasi-morphism} on the group is a function \(\mu :G\longrightarrow \mathbb{R}\) which satisfies \(\left| \mu (xy)-\mu (x)-\mu (y)\right| \leq B\) for every \(x,y\in G\) and a universal \(B\).NEWLINENEWLINELet \(H\) be a countable discrete set on the open interval \((0,1)\). On the set \(\mathcal{L}:=H\times \mathbb{Z}\) consider the metric \(d\) given as sum of the usual metrics on its components and the projection \(h:\mathcal{L}\longrightarrow \mathbb{Z}\). Denote by \(\mathcal{QI}\) the quasi-isometries of \((\mathcal{L},d)\) into itself and by \(\mathcal{QI}^{h}\) the quasi-isometries which satisfy \(\left| (h(f(x))-h(f(y)))-(h(x)-h(y))\right| \leq C(f)\) where \(C(f)\) depends only on \(f\).NEWLINENEWLINEA \textit{representation} of \(G\) is an injection from \(G\) to \(\mathcal{QI} ^{h}\). Denote by \(\mathrm{Rep}(G)\) the set of all representations. Furthermore, let \( \mathcal{PHQM}(G)\) denote the projective space of the linear space of homogeneous quasi-morphisms on \(G\). Then, the main result is the following (universal embedding for countable groups with nonzero quasi-morphism):NEWLINENEWLINETheorem. Let \(G\) be a countable group which carries at least one nontrivial homogeneous quasi-morphism. There exists an injection \(\mathcal{ PHQM}(G)\longrightarrow \mathrm{Rep}(G)\).