On the Bass-Lubotzky question about quotients of hyperbolic groups
The author proves the following Theorem 1. Every non-elementary hyperbolic group \(G\) has a non-trivial quotient \(Q=G/N\) which is finitely presented and which possesses no non-trivial finite homomorphic images. The result was asked by Bass and Lubotzky, who needed it to complete their construction of counter-examples to the Platonov conjecture. The proof of Theorem 1 uses techniques of generalized small cancellation which were developed in previous papers by the author. The author proves also Theorem 2. The following conditions are equivalent: (i) There exists a non-residually finite hyperbolic group. (ii) There exists a non-trivial hyperbolic group having no proper subgroups of finite indices. (iii) There exists a finite collection of non-elementary hyperbolic groups such that the set of the isomorphism classes of their common non-Abelian, finite, simple homomorphic images is finite. Whether statements (i), (ii) and (iii) are true is an open problem. In his paper [Publ., Math. Sci. Res. Inst. 8, 75-263 (1987; Zbl 0634.20015)], \textit{M. Gromov} conjectures that statement (ii) is true.
scientific article; zbMATH DE number 1448612