### On some free semigroups, generated by matrices

Let $$A=\left[\begin{array}{cc}1& 2\\ 0& 1\end{array}\right],\phantom{\rule{1.0em}{0ex}}{B}_{\lambda}=\left[\begin{array}{cc}1& 0\\ \lambda & 1\end{array}\right].$$ We call a complex number $\lambda $ “semigroup free“ if the semigroup generated by $A$ and ${B}_{\lambda}$ is free and “free” if the group generated by $A$ and ${B}_{\lambda}$ is free. First families of semigroup free $\lambda $’s were described by J. L. Brenner, A. Charnow (1978). In this paper we enlarge the set of known semigroup free $\lambda $’s. To do it, we use a new version of “Ping-Pong Lemma” for semigroups embeddable in groups. At the end we present most of the known results related to semigroup free and free numbers in a common picture....