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 ?
Definition 1 Let 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 ?
- 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 5 An infinite order element of is strongly contracting iff there exists such that no subpath of is labelled by a conjugate of .
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 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.