  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.
