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


  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 {<h>} 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.


About metric2011

metric2011 is a program of Centre Emile Borel, an activity of Institut Henri Poincaré, 11 rue Pierre et Marie Curie, 75005 Paris, France. See
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