Residual Finiteness of Outer Automorphism Groups of Finitely Generated Non-Triangle Fuchsian Groups

Abstract

Communicated by J. HowieIn [5] Grossman showed that outer automorphism groups of free groups and of fundamental groups of compact orientable surfaces are residually finite. In this paper we introduce the concept of "Property E" of groups and show that certain generalized free products and HNN extensions have this property. We deduce that the outer automorphism groups of finitely generated non-triangle Fuchsian groups are residually finite.

Residual finiteness of outer automorphism groups of certain 1-relator groups

“…To prove the lemma, we use Lemma 4.4 in [8]. Clearly (H2) and (H3) in Lemma 4.4 in [8] hold by above.…”

“…To prove the lemma, we use Lemma 4.4 in [8]. Clearly (H2) and (H3) in Lemma 4.4 in [8] hold by above. For (H1) in Lemma 4.4 in [8] Lemma 4.4 in [8], if α is a conjugating endomorphism of G such that α(a 1 ) = a 1 , then α(t) = a −s 1 ta s 1 for some s.…”

“…Definition 1.2 [2,8] . A group G has Property A (or E) if for each conjugating automorphism (or endomorphism, respectively) α of G, there exists a single element k ∈ G such that α(g) = k −1 gk for all g ∈ G, i.e., α = Inn k.…”

Class-Preserving Automorphisms of Certain HNN Extensions of Baumslag-Solitar Groups

“…This result has been used to study the residual finiteness of outer automorphism groups of certain groups. For example, outer automorphism groups of Fuchsian groups [1,20], most of Seifert 3-manifold groups [2] and certain 1-relator groups [16,17] are residually finite.…”

“…It is interesting to find some groups having Property A. Nontrivial free products of groups have Property A [1,22]. Generalized free products of finitely generated nilpotent groups, amalgamating an infinite cyclic subgroup, have Property A [29].…”