As part of my research, I have been trying to construct a spherical space at infinity for every right-angled artin group. I've been able to work it out for a certain class of defining graphs. I'd like to know how large this class is (I.e. does it consist of all graphs satisfying a certain invariant equation?), but I have very little graph theoretical background.

The class of graphs is invariant under disjoint union, join (connecting every vertex in one graph to every vertex in another), and the operation of identifying a single vertex in two disjoint graphs.

The "generating set" consists of all graphs not containing an induced square or an induced subgraph consisting of two triangles identified along a single edge (equivalently, it is the set of all graphs where every four-cycle bounds a tetrahedron). This set includes all trees and all graphs without 4-cycles, as well as all complete graphs.

My hope is that all or many hyperbolic 3-manifold groups virtually embed into RAAG's with such a defining graph. Some or all surface subgroups embed into such RAAG's (since any RAAG with a defining graph with an induced 5-cycle contains a surface subgroup). Thanks for your help!

Edit: I forgot to mention that RAAG's are subgroup separable iff their defining graphs have diameter at most 2 and contain no induced square.

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    $\begingroup$ Small correction: at best, you can hope for hyperbolic 3-manifolds to virtually embed into these RAAGs. $\endgroup$ – HJRW Mar 13 '13 at 15:57
  • $\begingroup$ As far as I know, there are very few known constraints on the RAAGs into which hyperbolic 3-manifolds virtually embed. $\endgroup$ – HJRW Mar 13 '13 at 15:58
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    $\begingroup$ In graph theory these are called C4,diamond-free graphs. $\endgroup$ – Benjamin Steinberg Mar 13 '13 at 16:22
  • $\begingroup$ (Deleted a slightly glib comment. I'm not sure that your operations buy you any new finite-volume hyperbolic 3-manifolds. Certainly free and amalgamated-over-cyclic products don't. I'm not certain about direct products (ie joins) but I'd be surprised if they did.) $\endgroup$ – HJRW Mar 13 '13 at 16:32
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    $\begingroup$ A related question I was curious about is which hyperbolic 3-manifold groups virtually embed into a hyperbolic right-angled Coxeter group. Most likely the sorts of cubulations produced by Kahn-Markovic/Bergeron-Wise would not be suitable for this sort of embedding. $\endgroup$ – Ian Agol Mar 13 '13 at 19:02

This is not a complete answer, but after tlking to some people at a comference about this, the only virtual embeddings we could think of explicitly are related to reflection groups.

Andreev's theorem (or Rivin's extension) characterizes right-angled hyperbolic polyhedra; in particular, they have trivalent boundary graphs (I think that the only other characteristic they have is that they lack prismatic circuits of size less than four). The reflection group is a right angled coxeter group whose defining graph is the dual graph of the polyhedron's boundary graph. There is a finite orbifold cover that is a manifold, and so these manifolds virtually embed into these right angled coxeter groups (one of the comments mentioned this kind of embedding).

Since the defining graphs are dual to trivalent graphs, they consist of triangulations of surfaces, which contain nothing but diamonds.

For my research, RACG's and RAAG's have identical subdivision rules, so diamonds may be more common than I thought.


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