Residually finite tubular groups

Abstract

A tubular group G is a finite graph of groups with Z 2 vertex groups and Z edge groups. We characterize residually finite tubular groups: G is residually finite if and only if its edge groups are separable. Methods are provided to determine if G is residually finite. When G has a single vertex group an algorithm is given to determine residual finiteness.

Tubular groups, 1-relator groups and nonpositive curvature

“…We note that the forthcoming paper [17] characterises exactly when a tubular group is residually finite.…”

Subgroups of direct products of graphs of groups with free abelian vertex groups

“…Despite their simple definition, they have a surprisingly rich source of diverse behaviour. Tubular groups provide examples of finitely generated 3-manifold groups that are not subgroup separable; of free-by-cyclic groups that do not act properly and semi-simply on a CAT(0) space; of groups that are CAT(0) but not Hopfian, etc (see [20] and references there). In the same paper, the authors give a characterisation of cyclic subgroup separable tubular groups and prove that a tubular group is cyclic subgroup separable if and only if it is residually finite if and only if it is virtually primitive.…”

“…In [20], the authors prove that for tubular groups the properties of being cyclic subgroup separable, of being residually finite and of having a finite index subgroup with isolated edge groups are equivalent. Furthermore, in [11], Button proves that a tubular group whose underlying graph is a tree is virtually special.…”

Автоморфизмы Группы Герстена

“…Так как группа G вкладывается в группы Aut(F n ), n ≥ 3, и Out(F n ), n ≥ 4, группы Aut(F n ), n ≥ 3, и Out(F n ), n ≥ 4, не могут действовать на односвязном геодезическом метрическом CAT(0)-пространстве собственно и кокомпактно изометриями. Группа Герстена и ее обобщения упоминаются в работах [3][4][5] и др.…”