A class of groups, called picture groups, is defined. Richard Thompson’s groups F , T , and V are all picture groups. Each picture group is shown to act properly and isometrically on a CAT(0) cubical complex. In particular, all picture groups are a-T-menable. Mathematics Subject Classifications (2000). 20F65, 43A15.

We prove the Borel Conjecture for a class of groups containing word-hyperbolic groups and groups acting properly, isometrically and cocompactly on a finite dimensional CAT(0)-space.

We exhibit 3-generator Artin groups which have finite 2-dimensional Eilenberg-Mac Lane spaces, but which do not act properly discontinuously by semi-simple isometries on a 2-dimensional CAT(0) complex. We prove that infinitely many of these groups are the fundamental groups of compact, non-positively curved 3-complexes. These examples show that the geometric dimension of a CAT(0) group may be s...

We exhibit 3-generator Artin groups which have finite 2-dimensional Eilenberg-Mac Lane spaces, but which do not act properly discontinuously by semi-simple isometries on a 2-dimensional CAT(0) complex. We prove that infinitely many of these groups are the fundamental groups of compact, non-positively curved 3-complexes. These examples show that the geometric dimension of a CAT(0) group may be s...

Any diagram group over a +nite semigroup presentation acts properly, freely, and cellularly by isometrices on a proper CAT(0) cubical complex. The existence of a proper, cellular action by isometries on a CAT(0) cubical complex has powerful consequences for the acting group G. One gets, for example, a proof that G satis+es the Baum–Connes conjecture. Any diagram group over a +nite presentation ...

We give a classification of irreducible characters of finite groups of Lie type of p′-degree, where p is any prime different from the defining characteristic, in terms of local data. More precisely, we give a classification in terms of data related to the normalizer of a suitable Levi subgroup, which in many cases coincides with the normalizer of a Sylow p-subgroup. The McKay conjecture asserts...

In this note we prove a weak hyperbolization conjecture for CAT(0) 3-dimensional Poincaré duality groups.

We prove a weak hyperbolization conjecture for CAT(0) 3-dimensional Poincaré duality groups.