The purpose of this post is to prepare for a discussion in our next post of three papers by Bridson, Howie, Miller and Short — the first about subgroups of direct products of surface groups, the second about subgroups of direct products of limit groups, and the third developing applications to residually free groups.
Definition 1. A group is residually free if for any , , there is a morphism such that .
Definition 2. A group is fully residually free (or -residually free) if for any finite set such that , there is a morphism such that .
Definition 3. A limit group is a finitely generated fully residually free group.
First examples of limit groups include free groups (of course), free abelian groups (not too hard to prove), and fundamental groups of closed hyperbolic surfaces with (see Wilton’s survey).
Equivalent definitions of limit groups
An alternative definition of a limit group, based on sequences of morphisms to free groups and their stable kernels, can be found in the surveys of Bestvina and Feighn and Wilton. A further definition, involving limits of finitely generated free groups in spaces of marked Cayley graphs, is described by Champetier and Guirardel.
As we will explain, starting with the examples of free groups, free abelian groups and surface groups, we can inductively obtain a large family of limit groups: the -residually free towers. They play a major role, on account of the following theorem, obtained independently by Kharlampovich and Myasnikov, and by Sela, which provides a more explicit way to look at limit groups.
Theorem 3. Limit groups are exactly the finitely generated subgroups of -residually free towers.
An -residually free tower is obtained as the fundamental group of a complex , for some . Such a complex is built in levels as follows:
- is a wedge of graphs, -dimensional tori , and closed hyperbolic surfaces (with ).
- is constructed by attaching to either of the following:
- A compact surface with boundary (hyperbolic, ), attached by the boundary, and with the condition that there must exist a retraction with non-abelian.
- A torus of any dimension , attached by a coordinate curve (i.e. ). It must also satisfy that the image of this coordinate curve in generates a maximal abelian subgroup of . (This amounts to saying that it is primitive and does not lie in a previous torus block.)
Key properties of limit groups
- Limit groups are torsion–free. (After all, residually free group are torsion free.)
- Limit groups are of type . (A consequence of the results in the paper by Bestvina and Feighn about constructible limit groups).
- A finitely generated subgroup of a limit group is a limit group itself. (It is again fully residually free.) So there are no dragons to be found amongst the subgroups of limit groups!
- Abelian subgroups of limit groups are finitely generated. (See Bestvina-Feighn.)
- Limit groups are commutative transitive: for all , if then (This is a particularly useful case of the next property).
- Limit groups are exactly the finitely generated groups that have the same one-quantifier first-order theory as a free group — specifically, for a non–abelian limit group, and for . (See Champetier-Guirardel.)
- A freely indecomposable limit group splits non–trivially as a graph of groups with cyclic edge stabilizers
Factoring homomorphisms and Makanin–Razborov diagrams
Limit groups come up in the study of for a finitely generated group , through the following theorem.
Theorem 4. Let be a finitely generated group. There exist a finite family of epimorphisms such that:
- Each is a limit group.
- Every morphism factors through some map in this family, i.e. for some , and some morphism .
The family of quotient maps in the theorem is called the (first layer of the) Makanin-Razborov diagram of . Let . It is easy to prove from the theorem that is residually free if and only if . Moreover, is the maximal residually free quotient of .
Consider the map . It is clear that is its kernel. So every residually free group embeds into a direct product of limit groups. This will allow us to apply theorems about subgroups of direct products of limit groups to study residually free groups. We will see examples of this in the next post.