This procedure transforms a group into another group which has the advantage of acting without fixed points on a 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 is infinite, then no , has Kazhdan’s Property (T).
We know from Novikov-Adyan that is infinite for 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 , a wall structure is easy: each wall is a set of edges whose interiors disconnect . 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.
Let be finitely generated by , . Let be the -homology cover of , i.e. its fundamental group is the kernel of .
When , has a canonical wall structure: take as walls preimages of edges of . By construction, their interiors separate . This also works for higher values of , after some preliminary work.
Lemma 2 There exists a group , containing a copy of , such that .
Indeed, any automorphism of lifts to . Let be the group generated by these lifts. We call it the cubization of .
Lemma 3 If is a word in , trivial in , then is trivial in .
Therefore, if , is a quotient of .
Lemma 4 If is infinite, has unbounded orbits on the set of walls of .
Indeed, let be an edge in . A geodesic joining to in passes at most once through an edge. Therefore its lift to crosses a large number of walls, hence an element achieves large, for the wall .
Note that the construction does not provide a proper action on Hilbert space. Not enough walls.