On the Bass–Lubotzky Question about Quotients of Hyperbolic Groups

We prove at Theorem 1 that any non-elementary hyperbolic group G possesses a non-trivial finitely presented quotient Q having no non-trivial subgroups of finite indices. The theorem was “commissioned” in August 1997 by H. Bass and A. Lubotzky because the statement was required to constructing of their counter-examples to Platonovʹs conjecture on the representation rigid linear groups [BL]. Theorem 2 gives three equivalent reformulations of the known problem on existence of non-residually finite hyperbolic groups.