Notes of David Hume’s Cambridge lecture 12-01-2017

Coarse negative curvature in action

1. Generalised loxodromics

Say an action of a group {G} on a hyperbolic metric space {X} is non-elementary acylindrical if for all {\epsilon>0}, there exist {R} and {N} such that for all points {x,y} with {d(x,y)>R}, the number of elements of {g} that move eaxh of {x} and {y} less than {\epsilon} away is less than {N}.

Say an element {g\in G} is a generalised loxodromic if there exists a non-elementary acylindrical action of {G} on some hyperbolic space {X} in which {g} acts loxodromically.

Here is one more equivalent definition of acylindrical hyperbolicity: say {G} is acylindrically hyperbolic if it admits a generalised loxodromic element.

2. Unexpected examples

Dahmani-Guirardel-Osin show that there exists {N} such that for every generalised loxodromic {g}, {\ll g^N \gg} is free. In particular, {G} is not simple.

Cantat-Lamy: the Cremona group of {P^2} is not simple. In fact, what they prove is that it is acylindrically hyperbolic.

Let {K} be a compact form of a real split (not Lie) Kac-Moody group. Then {K} is not simple (conjectured by Damour-Hillman, proved by Ghate-Horn-Köhl-Weiss). They show it has a Lie group quotient. Caprace and I add that {K} is acylindrically hyperbolic, as is the kernel of the morphism to a Lie group. We use actions on buildings.

Question. What is the relation between these actions and coarse negative curvature in {G} ?

3. Contraction

Definition 1 Let {Y} be a closed subset in a proper geodesic space {X}. Let {\rho(r)=o(r)} be some positive function. Say that {Y} is {\rho}-contracting if for all {k>0} and for all {z\in X} such that {d(z,Y)=r>k}, the diameter of the projection of {B(z,k)} is {\leq\rho(r)}.

For instance, in hyperbolic metric spaces, geodesics are {\rho}-contracting with {\rho} bounded. There, geodesics are Morse, but this holds in a wider generality.

Theorem 2 (Arzhantseva-Cashen-Gruber-Hume) A geodesic is Morse iff it is {\rho}-contracting for some {\rho=o(r)}.

Bestvina-Bromberg-Fujiwara, Osin: Let {G} be a fg group, {g\in G} of infinite order and undistorted. If the cyclic subgroup generated by {g} is contracting, then {g} is generalised loxodromic.

Sisto: If {g} is a generalised loxodromic, then {\langle g\rangle} is Morse in {G}.

Question. When can generalised loxodromics be understood via contraction ?

4. A program

I propose a program. Fix a popular group {G}.

  1. Does {G} admit a generalised loxodromic ?
  2. Can we classify the generalised loxodromics in {G}.
  3. Do generalised loxodromics become loxodromic in a unique action (call it universal), or does one need many ?

Examples

  • {G} amenable (or satisfies a law) has no generalised loxodromics.
  • {G} hyperbolic: generalised loxodromics coincide with elements of infinite order. Action on {G} itself is universal.
  • {G} acts geometrically on a {CAT(0)} space. A geodesic is Morse iff it is strongly contracting iff it is rank 1 (does not bound a half-flat). Generalised loxodromics coincide with elements translating an rank 1 geodesic. When is {G} universal ? Known: RAAG are universal.
  • {G=}MCG. Then generalised loxodromics coincide with pseudo-Anosov elements. The action on the curve complex is universal.
  • {G} relatively hyperbolic. Generalised loxodromic coincide with infinite order elements which are not conjugate to elements of {H}, plus all conjugates of generalised loxodromics of {H}. So if {H} has no generalised loxodromics, the action on the coned-off Cayley graph is universal. Universality is hard in general: Abbott showed that Dunwoody’s inaccessible group is not universal.

Theorem 3 (Abbott-Hume-Osin) Let {G} be hyperbolic rel. {H}. Let {H} act on {Y} nonelementarily acylindrically. Fix elements {h_i} of {H} acting loxodromically on {Y}. Then there exists a nonelementary acylindrical action of {G} on a hyperbolic space {X} such that every element {g} of infinite order not conjugate into {H} or conjugate to some {h_i} acts loxodromically.

5. Graphical small cancellation

Let {\Gamma} be a graph with oriented and {S}-labelled edges. Let {G(\Gamma)} be the group generated by {S\cup S^{1}} with relators all lables of closed loops in {\Gamma}. May be trivial. On makes small cancellation conditions, which

  • Ensure that {\Gamma} embeds in {Cay(G,S)}.
  • ensures that if {\Gamma} is finite, {G} is hyperbolic.

Theorem 4 (Arzhantseva-Cashen-Gruber-Hume) Let {G=G(\Gamma)}. Let {\gamma} be a geodesic in {Cay(G,S)}. Then {\gamma} is {\rho}-contracting iff every path in {\Gamma} labelled by a subword of {\gamma} is {\rho'}-contracting

This allows to understand the other degrees of contraction as well.

Corollary 5 An infinite order element {g} of {G(\Gamma)} is strongly contracting iff there exists {n} such that no subpath of {\Gamma} is labelled by a conjugate of {g^n}.

Corollary 6 There exist groups which are not subgroups of hyperbolic groups such that every nontrivial element is a generalized loxodromic (including some that copntain expanders).

With Gruber and Sisto, we prove that graphical small cancellation groups have generalized loxodromics.

With Abbot, we construct, for every unbounded {o(r)} function {\rho}, a group and an element in it which is {\rho}-contracting but not generalized loxodromic. Similarly, an element which is generalized loxodromic but not {\rho}-contracting.

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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