Notes of Yago Antolin’s Cambridge lecture 13-01-2017

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 {G/C} admits a total order invariant under the left action of {G}. If so, an order on {C} will produce an order on {G} for which each coset is convex.

Examples.

  • {{\mathbb Z}^n} is orderable and any maximal subgroup is relatively convex.
  • {G} is orderable iff {G} 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 {Homeo^+({\mathbb R})} 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 {\phi} be the function on simplicial paths defined by

\displaystyle  \begin{array}{rcl}  \phi=\#\textrm{ vertex raises }-\#\textrm{ vertex falls }+\#\textrm{ positive edges }-\#\textrm{ negative edges }. \end{array}

(this defines a quasimorphism on the free group). Then

\displaystyle  \begin{array}{rcl}  u\leq v \Leftrightarrow \phi(\textrm{reduced path from }u\textrm{ to }v)\geq 0 \end{array}

defines an invariant order on vertices of the tree.

The positive cone {\{g\,;\, g>1\}} 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 {A}, {B} be ordered groups, {C<A} and {C<B} a relatively convex subgroup. Then

  • {A} and {B} are relatively convex in {G=A\star_C B}.
  • If {C} is orderable, so is {G}.
  • If the orders of {C}, {G/A} and {G/B} are efficiently computable, the resulting order on {G} is efficiently computable.

3. Finding relatively convex subgroups

Burns-Hale: if every nontrivial f.g. subgroup of {G} maps onto a nontrivial orderable group, then {G} is orderable.

Say a group is locally indicable if every finitely generated subgroup maps onto {{\mathbb Z}}. Thus locally indicable groups are orderable.

Theorem 2 (Antolin-Dicks-Sunic) Let {G_0\leq G} be a subgroup. Assume that for every finite subset {X\subset G}, {\langle X\cup G_0\rangle/\ll G_9 \gg} maps onto a nontrivial orderable group. Then {G_0} is relatively convex in {G}.

Definition 3 Say {G} is {n}-indicable if either it is generated by less than {n} elements, or it maps onto {{\mathbb Z}^n}.

Examples. Free groups, orientable surface groups for all {n}. Locally residually torsion-free nilpotent groups for {n=2}.

RAAGs are not locally {n}-indicable.

Corollary 4 In a locally {n}-indicable group, each maximal {n-1}-generated subgroup is relatively convex.

This applies to maximal cyclic subgroups of free groups. Compare Louder-Wilton.

Definition 5 Say {G} is nasmof if {G} is torsion-free and all its nonabelian subgroups map onto {F_2}..

Theorem 6 (Antolin-Minasyan) RAAGs are nasmof.

In a nasmof group, for all {n}, every maximal {n}-generated abelian subgroup is relatively convex in {G}.

This applies to limit groups.

About metric2011

metric2011 is a program of Centre Emile Borel, an activity of Institut Henri Poincaré, 11 rue Pierre et Marie Curie, 75005 Paris, France. See http://www.math.ens.fr/metric2011/
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