On the profinite topology of right-angled Artin groups

Abstract

In the present work, we give necessary and sufficient conditions on the graph of a right-angled Artin group that determine whether the group is subgroup separable or not. Moreover, we investigate the profinite topology of F 2 × F 2 and we show that the profinite topology of the above group is strongly connected with the profinite topology of F 2 .

Limit groups are subgroup conjugacy separable

“…It is based on an idea of Michailova combined with observations of V. Metaftsis and E. Raptis [14]. Consider a finitely presented group H given by G. Higman [12] H = x 1 , x 2 , x 3 , x 4 | r 1 , r 2 , r 3 , r 4 , …”

“…We show here that a direct product of free groups of rank 2 is not subgroup conjugacy separable. It is based on an idea of Michailova combined with observations of V. Metaftsis and E. Raptis [14]. Consider a finitely presented group H given by G. Higman [12]…”

“…by Proposition 1 in[14] L H is closed in the profinite topology of F 4 × F ′ 4 , if and only if L H ∩ (F 4 × {1}) is closed in the profinite topology of (F 4 × {1}) or equivalently if and only if the group H = x 1 , x 2 , x 3 , x 4 | r 1 , r 2 , r 3 , r 4 is residually finite. However, G. Higman in[12] proved that H possesses no proper normal subgroups of finite index, so L H is not closed in the profinite topology ofF 4 × F ′ 4 .…”

On right-angled Artin groups without surface subgroups

“…• A(Γ) is subgroup separable (i.e. every finitely generated subgroup is closed in the profinite topology), if and only if Γ does not contain an induced C 4 or an induced P 4 [24], which happens exactly when every subgroup of A(Γ) is also a right-angled Artin group [11]. • A(Γ) is virtually a 3-manifold group, if and only if each connected component of Γ is a tree or a triangle [9,15].…”

Co-contractions of graphs and right-angled Artin groups

“…A./ is a virtually 3-manifold group, if and only if each connected component of is a tree or a triangle; see Droms [5] and Gordon [9] A./ is subgroup separable, if and only if no induced subgraph of is a square or a path of length 3 by Metaftsis and Raptis [16]. This happens if and only if every subgroup of A./ is also a right-angled Artin group, again by Droms [7].…”