Two-dimensional real triangle quasirepresentations of groups

Definition. By two-dimensional real triangle quasirepresentation of group $G$ we mean the mapping $\Phi$ of group $G$ into the group of two-dimensional real triangle matrices $T(2,R)$ such that if
$$\Phi (x)=\begin{pmatrix} \alpha(x) &\varphi(x) \\ 0 &\sigma(x) \end{pmatrix},$$
then: \begin{tabular}[t]{l} 1) $\alpha,\,\sigma$ are homomorphisms of group $G$ into $R^*$;
2) the set $\big\{\|\Phi(xy)-\Phi(x)\Phi(y)\|;\,x,y\in G\big\}$ is bounded. \end{tabular}
For brevity we shall call such mapping a quasirepresentation or a $(\alpha,\sigma)$-quasirepresentation for given diagonal matrix elements $\alpha$ and $\sigma$. We shall say that quasirepresentation is nontrivial if it is neither representation nor bounded. In this paper the criterion of existence of nontrivial $(\alpha,\sigma)$-quasirepresentation on groups is established. It is shown that if $G=A\ast B$ is the free product of finite nontrivial groups $A$ and $B$ and $A$ or $B$ has more than two elements then for every homomorphism $\alpha$ of group $G$ into $R^*$ there are $(\alpha,\varepsilon)$-, $(\varepsilon,\alpha)$- and $(\alpha,\alpha)$-quasirepresentation. Here the homomorphism $\varepsilon$ maps $G$ into 1.