14.6 (1999)

A group $\Gamma$ is said to have Property $P_{\text{nai}}$ if, for any finite subset $F$ of $\Gamma \setminus \{1\}$, there exists an element $y_0 \in \Gamma$ of infinite order such that, for each $x \in F$, the canonical epimorphism from the free product $\langle x \rangle * \langle y_0 \rangle$ onto the subgroup $\langle x, y_0 \rangle$ of $\Gamma$ generated by $x$ and $y_0$ is an isomorphism. For $n \in \{2, 3, \dots\}$, does $PSL_n(\mathbb{Z})$ have Property $P_{\text{nai}}$? More generally, if $\Gamma$ is a lattice in a connected real Lie group $G$ which is simple and with centre reduced to $\{1\}$, does $\Gamma$ have Property $P_{\text{nai}}$?