## Notes of Nicholas Touikan’s lecture

Hierarchical accessibility of relatively hyperbolic groups

joint with Lars Louder

Let ${\Gamma}$ be hyperbolic relative to some family P of parabolic subgroups. Assume ${\Gamma}$ is torsion free (non 2-torsion would suffice). Let ${E}$ be an elementary family of subgroups. ${E}$ could be ${\{}$trivial, ${{\mathbb Z}}$, subgroups conjugate into a parabolic subgroup${\}}$.

An ${(E-P)}$-splitting is a decomposition of ${\Gamma}$ as the fundamental group of a graph of groups such that edge groups are in ${E}$ and parabolic subgroups are conjugate into vertex groups. An ${(E,P)}$-hierarchy of ${\Gamma}$ is a rooted tree ${T(\Gamma)}$ that is grown, node by node, as follows.

0. The root is ${\Gamma}$.

1. If some node group ${\Gamma_a}$ in ${T(\Gamma)}$ is either free or admits no nontrivial ${(E,P)}$-splitting (say ${\Gamma_a}$ is ${(E,P)}$-rigid in this case), then stop.

1.1. If ${\Gamma_a}$ is not ${(E,P)}$-rigid, add as descendants the vertex groups of an ${(E,P)}$-splitting.

Statement (Delzant-Potyagailo).

If ${G}$ is a finitely presented and has no 2-torsion, then it admits a finite hierarchy.

It seems to us that there is a mistake in the proof, I explain what Lars and I have been able to repair.

Theorem 1 (Delzant-Potyagailo, Louder-Touikan) Every torsion free relatively hyperbolic group admits a finite canonical ${(E,P)}$-hierarchy such that

– If a node group ${\Gamma_a}$ is not one-ended, its descendants are the factors of a Grushko decomposition.

– If a node group ${\Gamma_a}$ is one-ended, its descendants are the vertex groups of its canonical ${(E,P)}$-JSJ decomposition (Guirardel).

Application: Haïssinsky-Carrasco have used our work to describe hyperbolic groups with low Ahlfors-regular conformal dimension.

1. Approach

From now on, I stick to the hyperbolic case (no parabolic subgroups).

We want a complex that expresses that vertex groups of a splitting are simpler, i.e. some complexity decreases. The difficulty is that there is no known complexity that does everything. For instance, ${F_2}$ can be viewed as an HNN extension of ${F_2}$, so rank does not decrease. So we have to resort to finiteness results.

Delzant-Potyagailo use orbihedra. Say a group ${\Gamma}$ acts on a simplicial complex ${C}$ without inversions if anytime an element stabilizes a simplex, it fixes all vertices. Such a group ${\Gamma}$, complex ${C}$ and action can be reconstructed from an orbihedron, the complex ${\pi=C/\Gamma}$, with stabilizers as face groups and inclusions, without forgetting the local model of the cover around vertices (two different orbihedra can have the same face groups but differ in local models).

Call volume of a orbihedron the number of simplices of ${\pi}$. Define triangular complexity of ${\Gamma}$ has the minimal volume of a orbihedron of which ${\Gamma}$ is the fundamental group.

Proposition 2 (Delzant-Potyagailo) Let ${\Gamma}$ ${E}$-act on a tree ${T}$. If for all separating vertices ${v}$, the vertex group ${\Gamma_v}$ acts elliptically on ${T}$, then there exists an other tree ${T'}$ with a surjective equivariant map ${T' \rightarrow T}$ so that we have an ${E}$-splitting ${T'/\Gamma =X}$ such that

$\displaystyle \begin{array}{rcl} T_E(\Gamma)\geq \sum_{v\mathrm{\,vertex\,in\,}X}T_E(X_v). \end{array}$

1.1. The problem

What if we have a chain ${\Gamma_i > \Gamma_{i+1} >...}$ where

– either ${\Gamma_i}$ is an HNN extension of ${\Gamma_{i+1}}$,

– or ${\Gamma_i}$ is obtained from ${\Gamma_{i+1}}$ by adding an ${n}$-th root of an element ?

Then triangular complexity does not decrease.

Assume that, eventually, all ${\Gamma_i}$ are one-ended. What will save the day is the following finiteness result.

Theorem 3 (Delzant, Sela, Reinfeldt-Weidman for hyp gps, Dahmani for rel hyp gps) Let ${G}$ be finitely presented, let ${\Gamma}$ by hyperbolic. If ${\psi_i : G \rightarrow \Gamma}$ are such that ${\psi_i(G)}$ are not elementary and non conjugate, then for some ${j}$, ${\psi_j}$ factors non trivially through an amalgamated product.