** Acylindrically hyperbolic groups from Kac-Moody groups **

Joint with Caprace.

Recall that a group is hyperbolically embedded in , denoted by , if there exists an isometric action of on some hyperbolic metric space such that

- is quasiconvex in .
- Tubular neighborhoods of orbits tend not to intersect much, i.e. for all ,
- acts properly on .

is acylindrically hyperbolic if it contains a free (non cyclic) hyperbolically embedded subgroup

**Examples**.

- Hyperbolic groups.
- Relatively hyperbolic groups.
- Mapping class groups. Use action on curve complex.
- . Use action on free factor complex.
- Cremona group. Use a perturbation of the action on infinite dimensional hyperbolic space.

Note that if where is vitually , then is not simple. Thus this gives a proof that Cremona group is not simple.

**1. Kac-Moody groups **

** 1.1. Definition **

These are built from copies of folowing a combinatorial recipe encoded in a generalized Cartan matrix such that

- ,
- ,
- .

**Example**. can be obtained in this manner. The matrix determines a (linear) graph which tells how to perform amalgamations.

If is a finite tree, the resulting group is an amalgam. This makes a rather weird looking group.

** 1.2. BN pairs **

Such groups come with a saturated twin BN pair , where is generated by and , as well as by and . is maximal abelian, is the finitely generated Coxeter group associated with the graph. and are Tits buildings of type . They are hyperbolic iff is hyperbolic. There is a Chevalley automorphism which swaps the roles of the two buildings, and on .

The orthogonal form of is the set of fixed points of . The Iwasawa decomposition states that .

**2. Results **

Theorem 1Let be a real Kac-Moody group with orthogonal form . Assume that is infinite, non virtually abelian, and not a non trivial direct product (almost always the case provided the matrix is complicated enough). Then there exists a virtually group which is hyperbolically embedded in .

I do not know wether the theorem is still true over finite fields.

** 2.1. Intermediate results **

Theorem 2Let be a group acting isometrically on a geodesic metric space . Let generate subgroup . Assume that

- projection onto some orbit is strongly contracting,
- is weakly properly discontinuous, i.e. for all and , there is such that only finitely many elements of satisfy .

Then there is a subgroup containing as a finite index subgroup and which is hyperbolically embedded in .

We use the following result of Caprace and Fujiwara. If a group acts on a building. If some element fixes an apartment and is rank 1, then has a strongly contracting orbit.

Theorem 3Let be a group acting isometrically on a building . Assume that chamber stabilizers are finite. Assume that some element acts as a hyperbolic isometry on with regular axis (far from any wall). Then is weakly properly discontinuous.

** 2.2. Proof of Theorem **

The orthogonal form satisfies all assumptions in previous theorems.