Left relatively convex subgroups

TL;DR

This paper introduces a criterion for left relative convexity of subgroups in groups, extending known theorems, and demonstrates that various important subgroups in different classes of groups are left relatively convex.

Contribution

It generalizes a theorem of Burns and Hale, providing new criteria and showing that maximal cyclic subgroups are left relatively convex in several classes of groups.

Findings

Maximal cyclic subgroups are left relatively convex in free, right-angled Artin, and certain surface groups.

Each free factor in a left orderable group is left relatively convex.

Edge groups in a graph of groups are left relatively convex in vertex groups under certain conditions.

Abstract

Let G be a group and H be a subgroup of G. We say that H is left relatively convex in G if the left G-set G/H has at least one G-invariant order; when G is left orderable, this holds if and only if H is convex in G under some left ordering of G. We give a criterion for H to be left relatively convex in G that generalizes a famous theorem of Burns and Hale and has essentially the same proof. We show that all maximal cyclic subgroups are left relatively convex in free groups, in right-angled Artin groups, and in surface groups that are not the Klein-bottle group. The free-group case extends a result of Duncan and Howie. We show that if G is left orderable, then each free factor of G is left relatively convex in G. More generally, for any graph of groups, if each edge group is left relatively convex in each of its vertex groups, then each vertex group is left relatively convex in the fundamental group; this generalizes a result of Chiswell. We show that all maximal cyclic subgroups in locally residually torsion-free nilpotent groups are left relatively convex.