Notes of Damian Osajda’s Cambridge lecture 12-01-2017

Group cubization

This procedure transforms a group {G} into another group {\tilde G} which has the advantage of acting without fixed points on a {CAT(0)} cubical complex. This is related to hyperbolization rather than cubulation, whence the word.

1. Burnside and Kazhdan

The main success of the construction is the following answer to a question of Shalom and Bekka-Dela Harpe-Valette.

Theorem 1 (Osajda) If free Burnside group {B(n,m)} is infinite, then no {B(n,km)}, {k>1} has Kazhdan’s Property (T).

We know from Novikov-Adyan that {B(n,m)} is infinite for {m} large, so this provides plenty of non-Kazhdan free Burnside groups.

To contradict Kazhdan’s property, it suffices to produce an action on a space with walls (Haglund-Paulin). For a graph {X}, a wall structure is easy: each wall is a set of edges whose interiors disconnect {X}. A group action on a space with walls automatically produces an affine isometric action on Hilbert space, existence of unbounded orbits passes from either side to the other.

2. Cubization

Let {G} be finitely generated by {S}, {\Gamma=Cay(G,S)}. Let {\tilde \Gamma} be the {{\mathbb Z}_k}-homology cover of {\Gamma}, i.e. its fundamental group is the kernel of {\pi_1(\Gamma)\rightarrow H_1(\Gamma,{\mathbb Z}_k)}.

When {k=2}, {\tilde \Gamma} has a canonical wall structure: take as walls preimages of edges of {\Gamma}. By construction, their interiors separate {\tilde \Gamma}. This also works for higher values of {k}, after some preliminary work.

Lemma 2 There exists a group {\tilde G}, containing a copy of {S}, such that {\tilde\Gamma=Cay(\tilde G,S)}.

Indeed, any automorphism of {\Gamma} lifts to {\tilde\Gamma}. Let {\tilde G} be the group generated by these lifts. We call it the cubization of {G}.

Lemma 3 If {w_G} is a word in {S}, trivial in {\Gamma}, then {w_{\tilde G}^k} is trivial in {\tilde G}.

Therefore, if {G=B(n,m)}, {\tilde G} is a quotient of {B(n,km)}.

Lemma 4 If {G} is infinite, {\tilde G} has unbounded orbits on the set of walls of {\tilde \Gamma}.

Indeed, let {e} be an edge in {\Gamma}. A geodesic joining {e} to {ge} in {\Gamma} passes at most once through an edge. Therefore its lift to {\tilde\Gamma} crosses a large number of walls, hence an element {\tilde g} achieves {d(\tilde w,\tilde g\tilde w)} large, for the wall {w=\pi^{-1}(e)}.

Note that the construction does not provide a proper action on Hilbert space. Not enough walls.

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