FUNDAMENTALNAYA
I PRIKLADNAYA MATEMATIKA
(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.
All articles are
published in Russian.
Location: http://mech.math.msu.su/~fpm/eng/95/954/95426.htm
Last modified: October 15, 1997.