I am wondering about the following generalization of the group $B_{1,2}=\langle a,b\, |\, bab^{-1}=a^2\rangle$: $$ G_k=\langle a_1,a_2,\ldots,a_{k+1}\, |\, a_{i+1}a_ia_{i+1}^{-1}=a_i^2, i=1,2,\ldots,k\rangle. $$ These groups, probably, were studied before; is there a name for these groups? One specific question:
Question: Are the groups $G_k$ linear?
The groups $G_k$ are not linear for $k>1$. This follows from the remark of Moishe Kohan. The relations $a_2a_1a_2^{-1}=a_1^2$ and $a_3a_2a_3^{-1}=a_2^2$ imply the relation $a^{2^{2^n}}_1=a^{n}_3a_2a^{−n}_3a_1a^{n}_3a^{−1}_2a^{−n}_3$. If we assume that $a_1,a_2,a_3$ are matrices, then the coefficients in $a^{2^{2^n}}_1$ grow at least as $2^{2^n}$, what is not possible for $a^n_3$.