** Coarse negative curvature in action **

**1. Generalised loxodromics **

Say an action of a group on a hyperbolic metric space is non-elementary acylindrical if for all , there exist and such that for all points with , the number of elements of that move eaxh of and less than away is less than .

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

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

**2. Unexpected examples **

Dahmani-Guirardel-Osin show that there exists such that for every generalised loxodromic , is free. In particular, is not simple.

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

Let be a compact form of a real split (not Lie) Kac-Moody group. Then 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 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 ?

**3. Contraction **

Definition 1Let be a closed subset in a proper geodesic space . Let be some positive function. Say that is -contracting if for all and for all such that , the diameter of the projection of is .

For instance, in hyperbolic metric spaces, geodesics are -contracting with 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 -contracting for some .

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

Sisto: If is a generalised loxodromic, then is Morse in .

**Question**. When can generalised loxodromics be understood via contraction ?

**4. A program **

I propose a program. Fix a popular group .

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

**Examples**

- amenable (or satisfies a law) has no generalised loxodromics.
- hyperbolic: generalised loxodromics coincide with elements of infinite order. Action on itself is universal.
- acts geometrically on a 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 universal ? Known: RAAG are universal.
- MCG. Then generalised loxodromics coincide with pseudo-Anosov elements. The action on the curve complex is universal.
- relatively hyperbolic. Generalised loxodromic coincide with infinite order elements which are not conjugate to elements of , plus all conjugates of generalised loxodromics of . So if 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 be hyperbolic rel. . Let act on nonelementarily acylindrically. Fix elements of acting loxodromically on . Then there exists a nonelementary acylindrical action of on a hyperbolic space such that every element of infinite order not conjugate into or conjugate to some acts loxodromically.

**5. Graphical small cancellation **

Let be a graph with oriented and -labelled edges. Let be the group generated by with relators all lables of closed loops in . May be trivial. On makes small cancellation conditions, which

- Ensure that embeds in .
- ensures that if is finite, is hyperbolic.

Theorem 4 (Arzhantseva-Cashen-Gruber-Hume)Let . Let be a geodesic in . Then is -contracting iff every path in labelled by a subword of is -contracting

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

Corollary 5An infinite order element of is strongly contracting iff there exists such that no subpath of is labelled by a conjugate of .

Corollary 6There 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 function , a group and an element in it which is -contracting but not generalized loxodromic. Similarly, an element which is generalized loxodromic but not -contracting.