Limit Groups Are Conjugacy Separable

Abstract

A limit group is a finitely-generated subgroup of a fully residually free group. We prove in this paper the result announced in the title.

Hereditary conjugacy separability of free products with amalgamation

“…Since limit groups are exactly virtual retracts of the latter class (see [23]) we deduce the main result of [4] as a combination of Propositions 2.7 and 2.15.…”

Magnus pairs in, and free conjugacy separability of, limit groups

“…This can be seen by cutting along the curves labelled u, v Lemma 2. 6 The homomorphism π 1 (U) → F 3 given by the continuous map in Fig. 2 Proof Lemma 2.6 and Proposition 2.5 imply that π 1 (U) is a Limit group.…”

“…This should be seen in contrast to the fact that limit groups are conjugacy separable [6]. Lioutikova [14], proved that iterated centralizer extensions (see Definition 4.3) of a free group F are freely conjugacy separable.…”

Limit groups are subgroup conjugacy separable

“…Note first that G is hereditarily conjugacy separable (see Proposition 3.8 in [7]). Let H be a finitely generated subgroup of G. By Theorem B [20] H is a virtual retract of G. Let K be a finitely generated subgroup such that K γ ≤ H for some γ ∈ G. We distinguish two cases.…”