# Examples of groups that are unknown to be acylindrically hyperbolic

Let $$G$$ be a group. We say that $$G$$ is acylindrically hyperbolic (for short, AH) if $$G$$ admits an isometric, acylindrical, and non-elementary action on some Gromov hyperbolic space $$X$$. Here is the Wikipedia page.

A lot of interesting groups are known to be AH; all non-elementary hyperbolic groups, the mapping class group of a closed surface, and Out($$F_n$$), the outer automorphism group of the free group of rank $$n$$.

Recently, the following groups are known to be AH.

1. Some Right-angled Artin groups (RAAGs)
2. Some fundamental groups of compact orientable 3-manifold
3. Some free-by-cyclic groups
4. Groups of deficiency at least 2

The purpose of the question is that I would like to know an example that is yet to be known as AH.

In particular, I was wondering if there exist weakly hyperbolic groups that are still unknown to be AH. The following is the definition of weakly hyperbolic groups.

Suppose a group $$G$$ acts isometrically on a Gromov hyperbolic space $$X$$. We say that $$G$$ is weakly hyperbolic if there exist two independent loxodromic elements, that is, two loxodromic elements such that their fixed point sets on the Gromov boundary $$\partial X$$ are disjoint.

I have seen that two dimensional Cremona group is AH but it is not known in the case the dimension is higher. Also, I was told that a virtually AH group is unknown to be AH. But I think that there are many other interesting groups that are unknown as AH.

Would you give such examples? Any comments (regardless of weak hyperbolicity) would be appreciated.

• I'm not sure if this is what you're asking, but there are groups that are weakly hyperbolic but not acylindrically hyperbolic: for any acylindrically hyperbolic group $G$, $G\times\mathbb{Z}$ is weakly hyperbolic but not acylindrically hyperbolic.
– HJRW
Jun 5 at 10:12
• Presumably it's unknown which (general) Artin groups are AH, since most things are unknown about general Artin groups. Jun 5 at 12:53
• @HJRW Actually, my main purpose is to know whether there is a group that is weakly hyperbolic but "unknown" to be AH but I have never thought about this. Thank you for answering! Jun 13 at 6:38
• @MattZaremsky Oh, right. I was told that many properties of the Artin-Tits groups are unknown. Thank you for your comment! Jun 13 at 6:39

Here is a list of groups for which acylindrical hyperbolicity is known or at least understood in some cases:

• The iconic examples are mapping class groups of non-sporadic surfaces of finite type and outer automorphism groups of free groups.
• (Infinitely presented) small cancellation groups. Here, small cancellation can refer to classical, graphical, or cubical small cancellation.
• Groups of deficiency two, i.e. groups admiting a presention of the form $$\langle x_1, \ldots, x_n \mid r_1, \ldots, r_m \rangle$$ with $$n \geq m+2$$.
• Cremona groups $$\mathrm{Bir}(\mathbb{P}^2_k)$$. There are also other groups from algebraic geometry known to be acylindrically hyperbolic, such as the automorphism group $$\mathrm{Aut}~k[x,y]$$, $$\mathrm{Tame}~ \mathrm{SL}_2(\mathbb{C})$$, and $$\mathrm{STame}(k^3)$$.
• Many 3-manifold groups.
• Irreducible graph products (including right-angled Artin/Coxeter groups).
• Many diagram groups.
• Irreducible graph braid groups.
• Higman's groups $$H_n:= \langle x_i \ (i \in \mathbb{Z}/n\mathbb{Z}) \mid x_{i+1}x_ix_{i+1}^{-1}= x_i^2 \rangle$$.
• Most free-by-cyclic groups. More precisely, $$\mathbb{F}_n \rtimes_\varphi \mathbb{Z}$$ is acylindrically hyperbolic if and only if $$\varphi$$ has infinite order in $$\mathrm{Out}(\mathbb{F}_n)$$. (Otherwise, the group is virtually $$\mathbb{F}_n \times \mathbb{Z}$$, so it cannot be acylindrically hyperbolic.)
• Most Coxeter groups.
• Partially CAT(-1) groups, i.e. groups acting geometrically on geodesically complete CAT(0) spaces with at least one point having a CAT(-1) neighbourhood.
• Many tubular groups. (A tubular group is the fundamental group of a graph of groups whose vertex-groups are $$\mathbb{Z}^2$$ and whose edge-groups are $$\mathbb{Z}$$.)
• Many Artin-Tits groups.
• Many automorphism groups of acylindrically hyperbolic groups, including hyperbolic groups, most relatively hyperbolic groups, multi-ended groups, graph products.
• Cactus groups.

With such a list, you can take an example, modify/generalise its definition a little bit, and ask whether it is acylindrically hyperbolic. So there are lots of groups for which we don't know whether they are acylindrically hyperbolic, but essentially because nobody took the time to work on the problem. Here are a few questions that should be more interesting:

• As mentioned in the comments, a well-known conjecture is that Artin-Tits groups are acylindrically hyperbolic, outside a few obvious cases where they are not.
• When are outer automorphism groups of right-angled Artin groups acylindrically hyperbolic? I solved the question for the automorphism group, but the problem for the outer automorphism is of different nature and is still open.
• Given a (say) torsion-free group $$G$$, is $$\langle G,x_1, \dots, x_n \mid w=1 \rangle$$ acylindrically hyperbolic if $$n \geq 2$$? This is related to the groups of deficiency two previously mentioned.
• (Asked by Minasyan-Osin) Which one-relator groups are acylindrically hyperbolic?
• (Asked by Minasyan-Osin) Is $$\mathrm{Aut}~k[x_1,\ldots, x_n]$$ acylindrically hyperbolic? As previously mentioned, a positive answer is known for $$n =2$$.
• As previously mentioned, one knows which free-by-cyclic groups are acylindrically hyperbolic. But what about free-by-free groups?
• Which surface braid groups are acylindricall hyperbolic? (I am not quite familiar with surface braid groups, so the question might not be relevant.)
• A question I find funny: Are CAT(0) groups with no $$\mathbb{Z}^2$$ acylindrically hyperbolic? This might not be a fair question since related to the rank-one rigidity conjecture.

Finally, I would like to mention a personal conjecture: (most) automorphism groups of finitely generated acylindrically hyperbolic groups are acylindrically hyperbolic themselves. As previously mentioned, the conjecture holds for hyperbolic groups, (most) relatively hyperbolic groups, multi-ended groups, graph products (including right-angled Artin/Coxeter groups).

About your question on weakly hyperbolic groups, there are many groups admitting an action of general type on a hyperbolic space but very far from being acylindrically hyperbolic. Here are a few examples:

• Acylindrically hyperbolic groups contain elements with virtually cyclic centralisers, so groups with an infinite center or splitting as a product of two infinite groups are not acylindrically hyperbolic. But many such examples admit non-trivial actions on hyperbolic spaces. For instance, product of hyperbolic groups or braid groups (they have infinite centers).
• In an acylindrically hyperbolic groups, an infinite normal subgroup must be acylindrically hyperbolic. So a group $$G$$ satisfying a short exact sequence $$1 \to N \to G \to Q \to 1$$ with $$N$$ infinite but not acylindrically hyperbolic, cannot be acylindrically hyperbolic itself. But, if $$Q$$ admits non-trivial actions on hyperbolic spaces, then so does $$G$$. For instance, cyclic extensions provide examples.
• More interesting examples are given by big mapping class groups. They are never acylindrically hyperbolic, but some of them act non-trivially on hyperbolic spaces.
• There are also some braided Thompson-like groups. They are never acylindrically hyperbolic (since they contain infinite normal (pure) braid groups with infinitely many strands, which are not acylindrically hyperbolic), but some of them act non-trivially on hyperbolic spaces (in fact, they are subgroups of big mapping class groups).
• Uniform lattices in products of unbounded hyperbolic spaces are never acylindrically hyperbolic, for instance because there are no cut-points in their asymptotic cones. Examples include the so-called BMW groups (for products of trees), such as the simple Burger-Mozes groups.

It is worth noticing that many of the groups mentioned above for which acylindrical hyperbolicity is unknown are "weakly hyperbolic". Free-by-free groups surjects onto free groups, so they act non-trivially on trees. Given a group of the form $$\langle G , x_1, \ldots, x_n \mid w=1 \rangle$$ with $$n \geq 2$$, you can quotient by the normal closures of $$G$$ or $$\langle x_1, \ldots, x_n \rangle$$ and see if you get a group acting on a hyperbolic space. Many Artin-Tits groups surjects onto free groups, and same thing for outer automorphism groups of right-angled Artin groups. Admitting an action of general type of some hyperbolic space is a property which is really weak, and there is no reason to think that it would be helpful to deduce some acylindrical hyperbolicity if you don't know something else about the action.

• I really appreciate your answer and many interesting examples! Indeed, I forgot to write one condition: I hope $G$ is weakly hyperbolic and the kernel of the action on the Gromov boundary $\partial X$ is finite (so this condition excludes the product case). But I think that this case is also solved by your comment! Thanks! Jun 20 at 7:03