## Notes of Emily Starck’s Cambridge lecture 12-01-2017

Commensurability classification of certain right-angled Coxeter groups et related surface amalgams

Joint with Pallevi Dani and Anne Thomas.

1. A subclass of RACG

We are interested in the subclass ${\mathcal{W}}$ of RACG associated to graphs ${\Gamma}$ such that

• no squares (${\Leftrightarrow}$ group is hyperbolic).
• connected, no separating clique (${\Leftrightarrow}$ 1-ended group).
• no triangles (${\Leftrightarrow}$ group is 2-dimensional).
• 3-convex: every topological edge is split in at least 3 combinatorial edges.
• with a cut par (${\Leftrightarrow}$ group splits over a 2-ended subgroup).
• no subdivided ${K_4}$ (${\Leftrightarrow}$ the JSJ decomposition of group contains a 2-ended and maximal hanging Fuchsian vertex group.

2. The class of geometric amalgams of free groups

Consider 2-dimensional hyperbolic ${P}$-manifolds, i.e. unions of ${\chi<0}$ surfaces glued along boundary components. ${\mathcal{G}}$ is the class of their fundamental groups. Their JSJ decompositions are bipartite graphs with vertex groups alternatively ${{\mathbb Z}}$ and surface.

The QI classification is known.

Theorem 1 (Malone 2010, Dani-Thomas 2014, Cachen-Martin 2016) Two elements of ${\mathcal{W}\cup\mathcal{G}}$ are QI iff they have isomorphic JSJ trees.

What about the commensurability classification ?

Theorem 2 (Dani-Starck-Thomas) Every element of ${\mathcal{W}}$ contains an element of ${\mathcal{G}}$ as a subgroup of finite index. Converse does not hold: there exists ${G\in\mathcal{G}}$ which is not QI to any group generated by torsion elements, so in particular to any element of ${\mathcal{W}}$.

3. Commensurability classification

We can do it for JSJ graphs of diameter ${\leq 4}$.

Diameter 2, in ${\mathcal{W}}$, corresponds to generalized ${\Theta}$ graphs. This correponds to a set of orbifolds glued along an edge. It has a vector ${v(W)}$ of Euler characteristics. Two examples ${W}$ and ${W'}$ are commensurable iff the vectors ${v(W)}$ and ${v(W')}$ are commensurable.

Diameter 4, in ${\mathcal{W}}$, corresponds to cycles of generalized ${\Theta}$ graphs. Two examples ${W}$ and ${W'}$ are commensurable iff either

1. the sets of commensurability classes of ${\Theta}$-graphs are the same for ${W}$ and ${W'}$ and a natural ratio holds.
2. there is one commensurability class of ${\Theta}$-graphs in each of ${W}$ and ${W'}$, and a natural ratio holds.

The heart of the proof consists of constructing finite covering spaces. We use Lafont’s topological rigidity theorem for 2 dimensional hyperbolic P-manifolds: in this class, isomorphic (resp. commensurable) fundamental groups implies homeomorphic (resp. homeomorphic finite covers).

Unfortunately, topological rigidity fails in the class of finite covers of the basic orbifold (corresponding to a cycle of 3 ${\Theta}$-graphs). We are led to thicken the singular locus.