Bristol University, May 12, 2008
Frédéric
Haglund, Paris–Sud
Groups acting on CAT(0) cube complexes
In his seminal paper on hyperbolic groups Gromov introduced CAT(0)
cube complexes as a source of non-positively curved spaces. In fact,
by the work of Chatterji–Niblo and Nica, many classical geometric
situations canonically lead to CAT(0) cube complexes.
I will give some general properties of groups acting properly on
CAT(0) cube complexes. Then I will study the nice case of right-angled
Coxeter groups. Lastly I will report on our joint work with Dani Wise:
when a group acts simply enough on a CAT(0) cube complex, the group
embeds in a right-angled Coxeter group. We say such a group is
special. On the one hand (virtually) special groups have excellent
algebraic properties. On the other hand surprisingly many groups are
virtually special, like arbitrary Coxeter groups and fundamental
groups of real hyperbolic compact manifold, provided they are
arithmetic of standard type.
Added May 13th: the principal paper on the material discussed in this talk is called "Special Cube Complexes" (coauthored with Dani Wise). It can be found here.