Notes of Jingyin Huang’s Cambridge lecture 11-01-2017

Groups quasi-isometric to right angled Artin groups

Joint with Bruce Kleiner.

1. Result

Counterexamples by Behrstock-Neumann, Ruiz-Kazachkov-Zakharov

Theorem 1 (Huang 2014) Let {G} and {H} be quasi-isometric right angled Artin groups. If {Out(G)} is finite, then {G} and {H} are commensurable.

Theorem 2 (Huang 2014) Let {G} and {H} be quasi-isometric f.g. groups. Assume {G} is a right angled Artin group, on a star-rigid graph (any automorphism fixing the star of a vertex pointwise is the identity) without induced 4-cycle. If {Out(G)} is finite, then {G} and {H} are commensurable.

Conversely, if {\Gamma} is a graph which contains an induced 4-cycle, this fails: there exists a f.g. generated group quasi-isometric but not commensurable to {G(\Gamma)}.

2. Proof

2.1. Analysis of quasi-isometries

The Salvetti complex {X} of a graph {\Gamma} consists of gluing tori, one for each clique of {\Gamma}. It is NPC (in fact, the cubical completion of the Cayley graph), it is a classifying sace for {G:=G(\Gamma)}.

We show that any quasi-isometry of {X} permutes maximal flats. Conversely, from a morphism of the space of flats of {X}, we are able to reconstruct a map {G\rightarrow G}. Furthermore, provided {Out(G)} is finite, every quasi-isometry is within finite distance of a flat preserving bijection (BKS 2007, Huang 2013).

2.2. Construction of an adapted space

Assume {H} is quasi-isometric to {G}. Then {H} acts by flat-preserving bijections. They may change the order of flats along standard geodesics. Add little branches to {X} in order separate flats from standard geodesics. One obtains a new {CAT(0)} space {Y} on which {H} acts isometrically. {Y} has a labelling and an orientation. If {\Gamma} is star-rigid, {H} preserves the labelling, but need not preserve the orientation. It does not suffice to go to a finite index subgroup.

2.3. Specialness

Say a NPC cube complex is special if it admits a local isometric embedding into some Salvetti complex. {Y/H} is a graph of spaces, where vertex spaces are again of the form {Y'/H'}. We prove a combination theorem for virtually special NPC cube complexes, inspired from Haglund-Wise 2009. There is a malnormality assumption, that Wise removed in his celebrated 2011 paper. In Haglund and Wise’s theory, hyperbolicity is essential. What saves us is that Salvetti complexes are better behaved than general NPC complexes: one may collapse bad tubes, get a hyperbolic cube complex and apply Haglund-Wise, and go back to the original space. Also, failure of malnormality is limited if no induced 4-cycles. Nevertheless, we need to modify Haglund-Wise’s canonical completion.

2.4. Counterexamples

If there are induced 4-cycles, the proof collapses. The construction of counterexamples involve replacing reducible lattices by irreducible lattices in products of trees.

3. Comments

Unclear wether star-rigidity is necessary.

Agol’s theorem (hyperbolic {CAT(0)}-cube complexes are virtually special) is more general than Wise’s. No assumption that the complex be a graph of spaces. Gives me hope.

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