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
- 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
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 .
Theorem 1 Let 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 2 Let 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 3 Let 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.