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Let $$A=\left[\begin{array}{cc}1& 2\\ 0& 1\end{array}\right],B_{\lambda }=\left[\begin{array}{cc}1& 0\\ \lambda & 1\end{array}\right].$$ 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....

We use notations: $[x,y]=[x_{,1}y]$ and $[x_{,k+1}y]$ e $[[x_{,k}y],y]$. We consider groups generated by $x$, $y$ satisfying relations $x=[x_{,n}y],y=[y_{,n}x]$ or $[x,y]=[x_{,n}y]$, $[y,x]=[y_{,n}x]$. We call groups of the first type mep-groups and of the second type nmep-groups. We show many properties and examples of mep- and nmep-groups. We prove that if $p$ is a prime then the group $S{l}_2(p)$ is a nmep-group. We give the necessary and sufficient conditions for metacyclic group to be a nmep-group and we show that nmep-groups with presentation $⟨x,y\mid [x,y]=[x_{,2}y],[y,x]=[y_{,2}x],x^n,y^m⟩$ are finite.