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Notes from talk 28 at NPCW01, Cambridge, January 13th 2017. A. Martin (UniversitÃ¤t Wien), joint work with A. Genevois.

\paragraph{Reminders about graph products} Let be a finite simplicial graph. For every , let be a group, and form the graph product

where denotes taking the normal closure.

When is discrete, this is a free product ; when is complete, a direct product.

If for all , (resp. ) then is a right angled Artin (resp. Coxeter) group.

\paragraph{Problem} Understand the structure and geometry of . Today’s “baby” case : is a long cycle (with length ). The strategy is to construct a curve complex , and an adequate action such that also acts on .

Recall that the Davis complex of a graph product is defined as follows:

- {vertices are cosets where is a clique of and denotes the graph product over .}
- {there is an edge from to whenever is a subclique of and .}
- {the cubes in the graph are filled.}

This yields a cube complex.

In our case, the Davis complex is the cubical subdivision of an -gonal complex . The stabilizers are as follows:

- {Stabilizers of vertices are conjugated to .}
- {Stabilizers of edges are conjugated to ‘s.}
- {Stabilizers of the whole polygon is trivial}

Strict fundamental domains are single polygons. Define a new complex from , as follows: start form a graph whose vertices represent the wall-trees, and edges intersections of wall-trees. Then fill the embedded -cycles.

is a complex.

Recall that is the small cancellation condition whereas means that the links of vertices have girth at least .

is *weakly acylindrical*. More precisely, write for the vertex of induced by a wall-tree of ; then, for any pait of vertices , in

- {If then .}
- {If then is a subgroup in a conjugate of a (Observe that those are not self-normalizing).}
- {If then is conjugated to a (Those are self-normalizing).}
- {If then and are conjugated to a .}

There exists an action .

*Proof:* First make act on ; then observe by the weak acylindricity property that the edges are preserved (via a discussion on the distance between the images of a pair of adjacent vertices).

The automorphism group of admits the following decomposition:

(Beware: Here denotes the automorphism group of labelled with the isomorphism classes of the ‘s.)

Assume that all the are finitely generated. Then is an acylindrical group.

Assume , where is a finite-dimensional and irreducible cube complex. Assume further that this action is essential, without fixed points at , and that one can find hyperplanes , such that is finite. Then is either virtually cyclic or acylindrically hyperbolic.

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