Geodesic
On some free semigroups, generated by matrices
Czechoslovak Mathematical Journal, Tome 65 (2015) no. 2, pp. 289-299 Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

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Let $$ A=\left [ \begin {matrix} 1 2 \\ 0 1 \end {matrix} \right ],\quad B_{\lambda }=\left [ \begin {matrix} 1 0 \\ \lambda 1 \end {matrix} \right ]. $$ 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.
Let $$ A=\left [ \begin {matrix} 1 2 \\ 0 1 \end {matrix} \right ],\quad B_{\lambda }=\left [ \begin {matrix} 1 0 \\ \lambda 1 \end {matrix} \right ]. $$ 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.
DOI : 10.1007/s10587-015-0175-4
Classification : 15A30, 20E05, 20M05
Keywords: free semigroup; semigroup of matrices 
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Słanina, Piotr. On some free semigroups, generated by matrices. Czechoslovak Mathematical Journal, Tome 65 (2015) no. 2, pp. 289-299. doi: 10.1007/s10587-015-0175-4

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