Tubular groups, 1-relator groups and nonpositive curvature

Abstract

We show (using results of Wise and of Woodhouse) that a tubular group is always virtually special (meaning that it has a finite index subgroup embedding in a RAAG) if the underlying graph is a tree. We also adapt Gardam and Woodhouse's argument on tubular groups which double cover 1-relator groups to show there exist 1-relator groups which are CAT(0) but not residually finite.

Finitely presented subgroups of direct products of graphs of groups with free abelian vertex groups

“…extend the results it suffices to show that groups in this class are CAT(0) and have unique roots. This has already been proven for tubular groups whose underlying graph is a tree in [3].…”

Residually finite tubular groups

“…This provided examples of nonautomatic one-relator groups that do not contain Baumslag-Solitar subgroups of the form BS(m, n) = a, t | (a m ) t = a n with m = ±n. Subsequently, Button observed that some of these one-relator groups are CAT(0) but not residually finite and has classified the residually finite snowflake groups [6]. We now reproduce his classification using our method.…”

“…groups with isoperimetric functions n d where d ∈ D is a dense subset of [2, ∞]. Gardam and Woodhouse showed that certain Snowflake groups embed as finite index subgroups of onerelator groups [9], and Button observed that many of these groups are not residually finite [6]. Cashen gave a quasi-isometric classification of tubular groups [7].…”

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.…”