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 of RACG associated to graphs such that
- no squares ( group is hyperbolic).
- connected, no separating clique ( 1-ended group).
- no triangles ( group is 2-dimensional).
- 3-convex: every topological edge is split in at least 3 combinatorial edges.
- with a cut par ( group splits over a 2-ended subgroup).
- no subdivided ( 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 -manifolds, i.e. unions of surfaces glued along boundary components. is the class of their fundamental groups. Their JSJ decompositions are bipartite graphs with vertex groups alternatively and surface.
The QI classification is known.
Theorem 1 (Malone 2010, Dani-Thomas 2014, Cachen-Martin 2016) Two elements of are QI iff they have isomorphic JSJ trees.
What about the commensurability classification ?
Theorem 2 (Dani-Starck-Thomas) Every element of contains an element of as a subgroup of finite index. Converse does not hold: there exists which is not QI to any group generated by torsion elements, so in particular to any element of .
3. Commensurability classification
We can do it for JSJ graphs of diameter .
Diameter 2, in , corresponds to generalized graphs. This correponds to a set of orbifolds glued along an edge. It has a vector of Euler characteristics. Two examples and are commensurable iff the vectors and are commensurable.
Diameter 4, in , corresponds to cycles of generalized graphs. Two examples and are commensurable iff either
- the sets of commensurability classes of -graphs are the same for and and a natural ratio holds.
- there is one commensurability class of -graphs in each of and , 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 -graphs). We are led to thicken the singular locus.