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 {<r^{n+1}}.

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}.

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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