JSJ I: History and Motivation

This is the first in a series of notes from a reading group on JSJ decompositions. This post is based on a talk given by Jonathan Fruchter.

 

Bass-Serre theory gives a way to cut up complicated groups into simpler pieces by considering their actions on trees. JSJ decompositions can be viewed as an extension of this philosophy which have proved to be very useful. Today we discuss the history and mention some of the applications of JSJ theory.

Groups acting freely on trees are known via Bass-Serre theory to be free. One can ask which groups act freely 0-hyperbolic metric spaces, since these are pretty close to being trees and are in fact known as $\mathbb{R}-$trees. One can observe that free abelian groups act on the real line freely by simply picking elements which are rationally independent and looking at the action of the group that they generate. Free groups act on trees, and less obviously surface groups do too: take a measured foliation, lift this to the universal cover, and contract every leaf of the foliation to a point. Using what is now known as the Rips machine, explained here, Rips proved that any group acting freely on an $\mathbb{R}-$tree has to be a free product of free abelian and surface groups, previously a conjecture of Morgan and Shalen. While much broader than the class of free groups, this is still very structured in some sense. Then people started wondering how much other structure one could find, and it turns out the answer is quite a lot. 

Theorem Let $G$ be a hyperbolic group acting on a real tree with no global fixed point and virtually cyclic segment stabilisers. Then $G$ splits as an amalgamated free product or HNN extension over a virtually cyclic subgroup.

Theorem Let $G$ be a non-rigid hyperbolic group. Then $G$ acts by isometries on a real tree $T$ with the following properties:

1. There is no point of $T$ fixed by all elements of $G$.
2. The stabiliser of every non-trivial segment in $T$ is virtually cyclic.

The proof idea is that if Out$(G)$ is infinite, we take a sequence of automorphisms $\{f_i\} \in \text{Aut}(G)$ representing different outer automorphism classes, which give rise to different actions of $G$ on its Cayley graph, and use these to construct an asymptotic cone. by taking the scaling limit $\frac{\text{Cay}G}{\text{"scaling factors"}}$. The scaling factors tend to $\infty$ so the limit is an $\mathbb{R}-$tree and the Rips machine gives rise to a small splitting. 

From these we can deduce

Theorem (Paulin) Let $G$ be a torsion-free non-abelian hyperbolic group. TFAE:

1. $G$ is non-rigid, i.e.Out$(G)$ is infinite.
2. $G$ acts on a real tree with no global fixed point and cyclic segment stabilisers
3. $G$ splits non-trivially as an amalgamated free product or HNN extension over $\mathbb{Z}$. (Here the image of $\mathbb{Z}$ must be root closed in the vertex groups).
   

1 $\implies$ 2 and $2 \implies 3$ are taken care of by the previous theorems so it suffices to exhibit outer automorphisms of infinite order in each of the cases covered by 3. By assumption $G$ is of the form $A_{*_1}, A_{*_\mathbb{Z}}, A_{*_1}B, A_{*_\mathbb{Z}}B$ where the factors are non-trivial. If $A$ were abelian, it would have to be cyclic since $G$ is hyperbolic and torsion-free. We could leave all of these as an exercise to the reader but will do one of them to be nice. 

We show that if $G=A_{*_\mathbb{Z}}B$, both non-abelian, then there is an outer automorphism of infinite order. The inspiration for this is the surface case: take a separating closed curve on a surface $S$, which is the cyclic group that the surface splits over and cuts the surface into components $A$ and $B$, and apply a self-homeomorphism to $A$ that fixes the boundary. This extends, via the identity, to a homeomorphism of the entire surface which is an infinite order element in the mapping class group of $S$. The general case $G=A_{*_\mathbb{Z}}B$ is similar: conjugate one of the factors by a generator of the $\mathbb{Z}$ subgroup and extend by the identity on the other factor. One checks that the factors have trivial centre, so the cyclic subgroup injects into the outer automorphism group.

Sela generalised this quite a lot. Since we are not Sela, here is a baby version of what he proved:

Lemma Let $G$ be a torsion-free non-abelian hyperbolic group. Then the following
assertions are equivalent.

1. $G$ is rigid, i.e. Out($G$) is finite.
2. For any hyperbolic group $H$, the set of conjugacy classes of monomorphisms $G \to H$ is finite.

Proof. $(2) \implies (1)$: Out($G$) is included in the set of conjugacy classes
of monomorphisms $G \to G$.

$(1) \implies (2)$ Assume that there is a hyperbolic group $H$ with infinitely many conjugacy classes of monomorphisms $G \to H$. The same construction which proves theorem 2 yields a sequence of distorted Cayley graphs of $H$ equipped with a $G$-action, with hyperbolicity constant converging to zero and hence an action of $G$ on a real tree with no global fixed point and cyclic segment stabilisers. Hence, Paulin's Theorem implies that $G$ is non-rigid. $\blacksquare$

The interested reader can leverage this lemma to prove

Theorem (Rips-Sela) Rigid torsion-free non-abelian hyperbolic group are co-Hopfian. 

The lemma also gives an algorithm to solve the isomorphism problem for rigid torsion-free hyperbolic groups: given two such grousp $G$ and $H$, enumerate monomorphisms from one to another and check if the compositions are inner.

A natural next question to ask is what happens when the groups aren't rigid. As often happens in GGT when people can't do the general case they specialise to a case with nicer geometry. In this case the nicer geometry means a hyperbolic 3-manifold. A theorem from the 1970's, called the JSJ decomposition theorem and is the namesake of general splitting theorems we are interested in, says that for nice $M$, there exist a finite collection of disjointly embedded tori $T_i$ such that every connected component of the complement is either atoroidal, which means that all tori are homotopic to the boundary, or Seifert-fibred, which means basically everywhere you look you find tori. This was previously discussed on the blog here.

As part of a broader programme to algebraise geometry, Kropholler started by showing that PD$^3$ groups with 'nice' centralisers admit an algebraic JSJ decomposition over virtually polycyclic groups. Sela improved upon this by proving

Theorem A 1-ended hyperbolic group $G$ has a $\mathbb{Z}-$splitting such that every vertex group is either

1. Rigid, i.e. it has no small splittings (this is the analogue of atoroidal), or
   
2. Quadratically hanging (this is the analogue of a surface with boundary, and adjacent edges correspond to boundary components of the surface)

This is proved using acylindrical actions on trees. Note that the edge groups can embed as powers of the boundary word.

Notice however that this is, at least in spirit, much closer the surface case than the 3-manifold case. We give a further analogy to illustrate this point. 

It is a famous fact that mapping class groups of surfaces are generated by Dehn twists about simple closed curves, and the Dehn-Nielsen-Baer theorem states that the extended mapping class group, i.e. where one allows orientation reversing homeomorphism, is isomorphic to the outer automorphism group of the surface. 

 Picture of a Dehn twist from Wikipedia

One can similarly define, for 1-ended hyperbolic groups $G$, a 'mapping class group' generated by a suitable notion of group-theoretic Dehn twists. Yet another theorem of Sela says

Theorem $[Out(G): Mod(G)]$ is finite for torsion-free $G$.

This gives a solution to the isomorphism problem for torsion-free hyperbolic groups which has been generalised in two directions: torsion-free toral relatively hyperbolic groups by Dahmani-Groves, and hyperbolic groups with torsion by Dahmani-Guirardel.

A JSJ decomposition for all finitely presented groups was obtained by Rips-Sela. The idea for the proof of the theorem on splittings of 1-ended hyperbolic groups is to view splittings as simple closed curves on a surface. If a curve $\gamma_2$ crosses the simple closed curve $gamma_1$ that we split over then the Dehn twist interacts with $\gamma_2$. Chasing this through in the analogy gives rise to two splitting $A_{*_C}B$ and $A'_{*_C'}B'$ being hyperbolic-hyperbolic, which means that the action of $C$ on the tree of the other splitting is hyperbolic and vice-versa. If the curves don't intersect, the Dehn twists do nothing, and the analogy is that the splitting subgroups act ellptically, so the splitting is called elliptic-elliptic. There are no hyperbolic-elliptic splittings. Hyperbolic-hyperbolic splittings give rise to QH subgroups via the Rips machine, and elliptic-elliptic splittings come from different parts of the JSJ.

Dunwoody gave examples to show that finite presentability is an essential assumption (we will hopefully cover this in a later post). Here are some further splitting results in this vein:

• Dunwoody-Sageev showed, by considering embedded graphs in the presentation 2-complex, that there is a JSJ for finitely presented groups over "slender" groups (all subgroups are finitely generated) under certain minimality assumptions of the splitting, e.g. splittings over $\mathbb{Z}^2$ are done under the assumption that there are no splittings over $\mathbb{Z}$.
• Fujiwara-Papasoglu showed, by considering actions on products of trees, that finitely presented groups admit a JSJ over slender groups
• Bowditch showed, by considering cut points in the boundary, that a 1-ended hyperbolic group $\Gamma$ admits a JSJ over 2-ended subgroups. Specifically, there is a finite graph-of-groups decomposition such that each edge group is virtually cyclic and each vertex group is either virtually cyclic, maximally hanging Fuchsian (MHF), or nonelementary and not MHF. No two vertex groups of the same type are adjacent, and each vertex group $G$ is full quasiconvex in $\Gamma$.
  

We close by mentioning that splittings are in some settings geometric: Papasoglu showed, among many other things, that splitting over 2-ended subgroups is a property preserved by quasi-isometries.