On Residualizing Homomorphisms Preserving Quasiconvexity^#

ABSTRACT

H is called a G-subgroup of a hyperbolic group G if for any finite subset M ⊂ G there exists a homomorphism from G onto a non-elementary hyperbolic group G ₁ that is surjective on H and injective on M. In his paper in 1993 A. Ol'shanskii gave a description of all G-subgroups in any given non-elementary hyperbolic group G. Here we show that for the same class of G-subgroups the finiteness assumption on M (under certain natural conditions) can be replaced by an assumption of quasiconvexity.

ACKNOWLEDGMENT

The author is grateful to Professor A. Yu. Ol'shanskii for suggesting the problem and helpful remarks.