(FUNDAMENTAL AND APPLIED MATHEMATICS)

1995, VOLUME 1, NUMBER 4, PAGES 1129-1132

## Two-dimensional real triangle quasirepresentations of groups

V.A.Faiziev

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:
1) $\alpha, \sigma$ are homomorphisms of group G into R*;
2) the set $\{\| \Phi(xy) - \Phi(x)\Phi(y)\|; x,y \in G \}$ is bounded.

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*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.

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