Divergence, thick groups and Morse geodesics
Definition 1 Fix constants and . Let X be a geodesic metric space, one ended. Given a triple of points in with , define
The divergence of is a function of .
Only its growth type matters.
Motivating example (Gromov): Let be a geodesic. Put , , , , . is then called the divergence of .
Example 1 Euclidean space has linear divergence.
More examples with linear divergence:
- Products of arbitrary infinite groups.
- Groups whose center contains an element of infinite order.
- Groups which satisfy a law.
- Fundamental groups of Seifert fiber spaces.
Example 2 Hyperbolic space has exponential divergence.
More examples with exponential divergence:
- Fundamental groups of hyperbolic manifolds,
- Hyperbolic groups (this one of the possible definitions of -hyperbolicity for one ended groups).
- Fundamental groups of negatively curved manifolds (with finite volume).
Conjecture. (Gromov, Asymptotic invariants…) For spaces, one expects geodesic rays issuing from a point to either diverge linearly or exponentially.
Theorem 2 (Gersten 1995) This fails. There exist 2-complexes with quadratic divergence.
In fact, quadratic divergence examples are abundant and natural. Gersten classified 3-dimensional graph manifold groups in terms of divergence, and quadratic divergence is frequent.
Theorem 3 (Behrstock, Duchin-Rafi) Mapping class groups have quadratic divergence.
Teichmüller spaces of high enough complexity (either in Weil-Peterson or Teichmüller metric) have quadratic divergence.
Most right-angled Artin groups have quadratic divergence.
Conjecture: has quadratic divergence. One knows that divergence is at most quadratic, superlinear. Outer space has at least quadratic divergence.
Question (Gersten): Is polynomial divergence possible on a space ?
Theorem 4 (Behrstock-Drutu, Macura) For any integer , there exists a space with polynomial divergence of degree . In fact, such examples can be constructed which admit a cocompact isometric action.
Next, I explain this construction.
We use the notion of thickness I developped with Cornelia Drutu and Lee Mosher.
Definition 5 A finitely generated group is strongly algebraically thick of degree 0 if it has linear divergence. A group is thick of degree at most if it contains a collection of quasiconvex subgroups satisfying
1. The union of these subgroups generates a finite index subgroup of .
2. Each in the familly is thick of order at most .
3. For each pair , , there is a sequence such that for each between 1 and , the intersection of and is infinite and coarsely path connected.
Motivating example: The mapping class group of a surface of genus with punctures is thick of degree at most 1 as long as complexity . Indeed, it is not of order 0 (superlinear divergence). But admits a nice generating set (Humphrey generators): Dehn twists along closed curves. Take to be generated by two independent Dehn twists. These are quasiconvex subgroups. By connectivity of the curve complex, axiom 3 is satisfied.
More examples of groups which are thick of order 1: Lots of Artin groups. . Graphmanifold groups.
Theorem 6 (Behrstock-Drutu) If a geodesic metric space is thick of order , then the divergence of is .
This how we get upper bounds on divergence. I do not know any other general method. Lower bounds are obtained by considering explicit examples of geodesics.
Theorem 7 For all , there exists an infinite family of non pairwise quasiisometric groups which are each
2. Thick of order .
3. With divergence of order .