On almost representations of groups

Abstract:

We say that a group $G$ belongs to the class $\mathcal {K}$ if every nonunit quotient group of $G$ has an element of order two. Let $H$ be a Hilbert space and let $U(H)$ be its group of unitary operators. Suppose that groups $A$ and $B$ belong to the class $\mathcal {K}$ and the order of $B$ is more than two. Then the free product $G=A\ast B$ has the following property. For any $\varepsilon >0$ there exists a mapping $T:G \to U(H)$ satisfying the following conditions : 1) $\Vert T(xy) - T(x)\cdot T(y) \Vert \le \varepsilon , \quad \forall x, \forall y \in G;$ 2) for any representation $\pi : G\to U(H)$ the relation \begin{equation*}\sup \{\Vert T(x) - \pi (x) \Vert ,x\in G\} =2\end{equation*} holds.