Expression de la differ\'entielle $d_3$ de la suite spectrale de Hochishild-Serre en cohomologie born\'ee r\'eelle

For discrete groups, we construct two bounded cohomology classes with coefficients in the second space of the reduced real ${\mathrm{\ell }}_1$-homology. Precisely, we associate to any discrete group $G$ a bounded cohomology class of degree two noted ${\mathfrak{g}}_2\in H_b^2(G,{\bar{H}}_2^{{\mathrm{\ell }}_1}(G,\mathbb{R}))$. For $G$ and $\mathrm{\Pi }$ groups and $\theta :\mathrm{\Pi }\to Out(G)$ any homomorphism we associate a bounded cohomology class of degree three noted $[\theta]\in H_b^3(\Pi, \bar{H}_2^{\ell_1}(G, \mathbb R))$\mathrm{.}Whentheouterhomomorphism$\theta : \Pi\rightarrow Out(G)$inducesanextensionof$G$by$\Pi$ we show that the class ${\mathfrak{g}}_2$ is $\mathrm{\Pi }$-invariant and that the differential $d_3$ of Hochschild-Serre spectral sequence sends the class ${\mathfrak{g}}_2$ on the class $[\theta ]$ : $d_3({\mathfrak{g}}_2)=[\theta ]$. Moreover, we show that for any integer $n\ge 0$ the differential $d_3:E_3^{n,2}\to E_3^{n+3,0}$ of Hochschild-Serre spectral sequence in real bounded cohomology is given as a cup-product by the class $[\theta ]$.