## Notes of Jason Behrstock’s lecture

Divergence, thick groups and Morse geodesics

1. Divergence

Definition 1 Fix constants ${\delta>0}$ and ${\gamma>0}$. Let X be a geodesic metric space, one ended. Given a triple of points ${a, b, c}$ in ${X}$ with ${d(c,\{a,b\})=r>0}$, define

$\displaystyle \begin{array}{rcl} div(a,b,c)=\textrm{distance between }a\textrm{ and }b\textrm{ in the complement of }B(c,\delta r-\gamma). \end{array}$

The divergence of ${X}$ is a function of ${n}$.

$\displaystyle \begin{array}{rcl} div_X(n)=sup_{\{(a,b,c)\, ; \,d(a,b)=n\}} div(a,b,c). \end{array}$

Only its growth type matters.

Motivating example (Gromov): Let ${g}$ be a geodesic. Put ${\delta=1}$, ${\gamma=0}$, ${c=g(0)}$, ${a=g(r)}$, ${b=g(-r)}$. ${div(a,b,c)}$ is then called the divergence of ${g}$.

Example 1 Euclidean space has linear divergence.

More examples with linear divergence:

• ${{\mathbb Z}^n}$,
• 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 ${\delta}$-hyperbolicity for one ended groups).
• Fundamental groups of negatively curved manifolds (with finite volume).

Conjecture. (Gromov, Asymptotic invariants…) For ${CAT(0)}$ spaces, one expects geodesic rays issuing from a point ${x_0}$ to either diverge linearly or exponentially.

Theorem 2 (Gersten 1995) This fails. There exist ${CAT(0)}$ 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: ${Out(F_n)}$ 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 ${CAT(0)}$ space ?

Theorem 4 (Behrstock-Drutu, Macura) For any integer ${n>0}$, there exists a ${CAT(0)}$ space with polynomial divergence of degree ${n}$. In fact, such examples can be constructed which admit a cocompact isometric action.

Next, I explain this construction.

2. Thickness

We use the notion of thickness I developped with Cornelia Drutu and Lee Mosher.

Definition 5 A finitely generated group ${G}$ is strongly algebraically thick of degree 0 if it has linear divergence. A group is thick of degree at most ${n+1}$ if it contains a collection of quasiconvex subgroups satisfying

1. The union of these subgroups generates a finite index subgroup of ${G}$.

2. Each ${H}$ in the familly is thick of order at most ${n}$.

3. For each pair ${H}$, ${H'}$, there is a sequence ${H = H_1, H_2,...,H_k=H'}$ such that for each ${i}$ between 1 and ${k}$, the intersection of ${H_{i-1}}$ and ${H_i}$ is infinite and coarsely path connected.

Motivating example: The mapping class group of a surface of genus ${g}$ with ${p}$ punctures is thick of degree at most 1 as long as complexity ${3g+p-3 \geq 2}$. Indeed, it is not of order 0 (superlinear divergence). But ${G}$ admits a nice generating set (Humphrey generators): Dehn twists along ${2g+1}$ closed curves. Take ${H}$ 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. ${Out(F_n)}$. Graphmanifold groups.

Theorem 6 (Behrstock-Drutu) If a geodesic metric space ${X}$ is thick of order ${n}$, then the divergence of ${X}$ 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 ${n}$, there exists an infinite family of non pairwise quasiisometric groups which are each

1. ${CAT(0)}$.

2. Thick of order ${n}$.

3. With divergence of order ${n+1}$.