The equivalence of some residual properties of word-hyperbolic groups
We recall that a group \(G\) is said to be residually finite if for each non-trivial element \(g\in G\), there is a finite quotient \(G\to\overline G\) such that the image of \(g\) in \(\overline G\) is non-trivial. Important open questions on word-hyperbolic groups are whether each such group is residually finite, and whether each such group is linear. (By a result of Mal'cev, an affirmative answer to the second question implies an affirmative answer to the first one.) The main result of this paper is the following Theorem. The following are equivalent: (a) Every word-hyperbolic group is residually finite. (b) Every non-trivial word-hyperbolic group has a non-trivial finite quotient.
scientific article; zbMATH DE number 1423886
scientific article