Convex subgroups of orderable groups
Could have been done 40 years ago.
Orderable means there exists a left-invariant total order. A subgroup is relatively convex if the set of cosets admits a total order invariant under the left action of . If so, an order on will produce an order on for which each coset is convex.
- is orderable and any maximal subgroup is relatively convex.
- is orderable iff acts faithfully on a totally ordered set by order preserving transformations.
- Free groups are orderable, but not in any canonical way. Here is the most natural construction (Ghys): two generic elements of generate a free group. Note the orderability of the free group was the key ingredient in Mineyev’s solution of Hanna Neumann’s conjecture. Note this conjecture extends to free products of orderable groups (Antolin-Martino-Schanesbrow).
1. Ordering trees
Dicks-Sunic started it.
Consider the Cayley tree of a free group. Define an order on links of vertices in an invariant manner. This allows define wether an oriented path occurs a vertex raise or fall at a vertex. Also, orient edges.
Let be the function on simplicial paths defined by
(this defines a quasimorphism on the free group). Then
defines an invariant order on vertices of the tree.
The positive cone of this order turns out to be a context-free language. A theorem by Hermillet-Sunic asserts that there is no better way to get such an order on the free group.
This extends to free products.
2. Ordering amalgamated products
Theorem 1 (Antolin-Dicks-Sunic) Let , be ordered groups, and a relatively convex subgroup. Then
- and are relatively convex in .
- If is orderable, so is .
- If the orders of , and are efficiently computable, the resulting order on is efficiently computable.
3. Finding relatively convex subgroups
Burns-Hale: if every nontrivial f.g. subgroup of maps onto a nontrivial orderable group, then is orderable.
Say a group is locally indicable if every finitely generated subgroup maps onto . Thus locally indicable groups are orderable.
Theorem 2 (Antolin-Dicks-Sunic) Let be a subgroup. Assume that for every finite subset , maps onto a nontrivial orderable group. Then is relatively convex in .
Definition 3 Say is -indicable if either it is generated by less than elements, or it maps onto .
Examples. Free groups, orientable surface groups for all . Locally residually torsion-free nilpotent groups for .
RAAGs are not locally -indicable.
Corollary 4 In a locally -indicable group, each maximal -generated subgroup is relatively convex.
This applies to maximal cyclic subgroups of free groups. Compare Louder-Wilton.
Definition 5 Say is nasmof if is torsion-free and all its nonabelian subgroups map onto ..
Theorem 6 (Antolin-Minasyan) RAAGs are nasmof.
In a nasmof group, for all , every maximal -generated abelian subgroup is relatively convex in .
This applies to limit groups.