## 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.