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.