## Notes of David Hume’s lecture

Acylindrically hyperbolic groups from Kac-Moody groups

Joint with Caprace.

Recall that a group ${H}$ is hyperbolically embedded in ${G}$, denoted by ${H\hookrightarrow_h G}$, if there exists an isometric action of ${G}$ on some hyperbolic metric space ${X}$ such that

1. ${H}$ is quasiconvex in ${X}$.
2. Tubular neighborhoods of orbits tend not to intersect much, i.e. for all ${g\notin H}$,

$\displaystyle \begin{array}{rcl} \mathrm{diameter}((H+R)\cap(gH+R))\leq K(R). \end{array}$

3. ${H}$ acts properly on ${X}$.

${G}$ is acylindrically hyperbolic if it contains a free (non cyclic) hyperbolically embedded subgroup

Examples.

1. Hyperbolic groups.
2. Relatively hyperbolic groups.
3. Mapping class groups. Use action on curve complex.
4. ${Out(F_n)}$. Use action on free factor complex.
5. Cremona group. Use a perturbation of the action on infinite dimensional hyperbolic space.

Note that if ${H\hookrightarrow_h G}$ where ${H}$ is vitually ${{\mathbb Z}}$, then ${G}$ 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 ${Sl(2,{\mathbb R})}$ folowing a combinatorial recipe encoded in a generalized Cartan matrix ${A\in M_n({\mathbb Z})}$ such that

1. ${a_{ii}=2}$,
2. ${a_{ij}\leq 0}$,
3. ${a_{ij}=0\Leftrightarrow a_{ji}=0}$.

Example. ${Sl(n,{\mathbb R})}$ can be obtained in this manner. The matrix determines a (linear) graph which tells how to perform amalgamations.

If ${T}$ is a finite tree, the resulting group ${G_T}$ is an amalgam. This makes a rather weird looking group.

1.2. BN pairs

Such groups come with a saturated twin BN pair ${(B_+,B_-,N)}$, where ${G}$ is generated by ${B_+}$ and ${N}$, as well as by ${B_-}$ and ${N}$. ${B_+\cap B_-=T}$ is maximal abelian, ${W=N/T}$ is the finitely generated Coxeter group associated with the graph. ${\Delta_+=G/B_+}$ and ${\Delta_-=G/B_-}$ are Tits buildings of type ${(W,S)}$. They are hyperbolic iff ${W}$ is hyperbolic. There is a Chevalley automorphism ${\sigma}$ which swaps the roles of the two buildings, and ${\sigma(t)=t^{-1}}$ on ${T}$.

The orthogonal form of ${G}$ is the set of fixed points of ${\sigma}$. The Iwasawa decomposition states that ${G=KB_+=KB_-}$.

2. Results

Theorem 1 Let ${G}$ be a real Kac-Moody group with orthogonal form ${K}$. Assume that ${W}$ 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 ${{\mathbb Z}}$ group ${H}$ which is hyperbolically embedded in ${G}$.

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

2.1. Intermediate results

Theorem 2 Let ${G}$ be a group acting isometrically on a geodesic metric space ${X}$. Let ${h\in G}$ generate subgroup ${H}$. Assume that

1. projection onto some orbit is strongly contracting,
2. ${h}$ is weakly properly discontinuous, i.e. for all ${D}$ and ${x\in X}$, there is ${M}$ such that only finitely many elements ${g}$ of ${G}$ satisfy ${d(x,gx)\leq Dd(h^Mx,gh^Mx)}$.

Then there is a subgroup ${H'}$ containing ${H}$ as a finite index subgroup and which is hyperbolically embedded in ${G}$.

We use the following result of Caprace and Fujiwara. If a group acts on a ${CAT(0)}$ building. If some element ${h\in G}$ fixes an apartment ${\mathcal{A}}$ and ${h_{|\mathcal{A}}}$ is rank 1, then ${}$ has a strongly contracting orbit.

Theorem 3 Let ${G}$ be a group acting isometrically on a ${CAT(0)}$ building ${X}$. Assume that chamber stabilizers are finite. Assume that some element ${h\in G}$ acts as a hyperbolic isometry on ${X}$ with regular axis (far from any wall). Then ${h}$ is weakly properly discontinuous.

2.2. Proof of Theorem

The orthogonal form ${K}$ satisfies all assumptions in previous theorems.