** Group cubization **

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.

**2. Cubization **

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.

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## About metric2011

metric2011 is a program of Centre Emile Borel, an activity of Institut Henri Poincaré, 11 rue Pierre et Marie Curie, 75005 Paris, France. See
http://www.math.ens.fr/metric2011/