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 Fundam. Prikl. Mat.: Year: Volume: Issue: Page: Find

 Fundam. Prikl. Mat., 1995, Volume 1, Issue 4, Pages 1129–1132 (Mi fpm111)

Short communications

Two-dimensional real triangle quasirepresentations of groups

V. A. Faiziev

Abstract: 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 $\{\|\Phi(xy)-\Phi(x)\Phi(y)\|; x,y\in G\}$ 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.

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Bibliographic databases:
UDC: 519.46

Citation: V. A. Faiziev, “Two-dimensional real triangle quasirepresentations of groups”, Fundam. Prikl. Mat., 1:4 (1995), 1129–1132

Citation in format AMSBIB
\Bibitem{Fai95} \by V.~A.~Faiziev \paper Two-dimensional real triangle quasirepresentations of groups \jour Fundam. Prikl. Mat. \yr 1995 \vol 1 \issue 4 \pages 1129--1132 \mathnet{http://mi.mathnet.ru/fpm111} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=1791800} \zmath{https://zbmath.org/?q=an:0867.20007}