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.

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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 http://www.math.ens.fr/metric2011/
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