Cohomology of Split Group Extensions and Characteristic Classes

There are characteristic classes that are the obstructions to the vanishing of the differentials in the Lyndon-Hochischild-Serre spectral sequence of an extension of an integral lattice L by a group G. These characteristic classes exist in a given page of the spectral sequence provided the differentials in the previous pages are all zero. When L decomposes into a sum of G-sublattices, we show that there are defining relations between the characteristic classes of L and the characteristic classes of its summands.Comment: 13 page

Crystallographic actions on contractible algebraic manifolds

We study properly discontinuous and cocompact actions of a discrete subgroup $\Gamma$ of an algebraic group $G$ on a contractible algebraic manifold $X$. We suppose that this action comes from an algebraic action of $G$ on $X$ such that a maximal reductive subgroup of $G$ fixes a point. When the real rank of any simple subgroup of $G$ is at most one or the dimension of $X$ is at most three, we show that $\Gamma$ is virtually polycyclic. When $\Gamma$ is virtually polycyclic, we show that $\Gamma$ is virtually polycyclic. When $\Gamma$ is virtually polycyclic, we show that the action reduces to a NIL-affine crystallographic action. As applications, we prove that the generalized Auslander conjecture for NIL-affine actions holds up to dimension six and give a new proof of the fact that every virtually polycyclic group admits a NIL-affine crystallographic action.Comment: This final version has been accepted for publication in 2013. The statements of the main results are now more general as they cover algebraic groups G where the real rank of any simple subgroup of G is at most on

Decomposing groups by codimension-1 subgroups

The paper is concerned with Kropholler's conjecture on splitting a finitely generated group over a codimension-1 subgroup. For a subgroup H of a group G, we define the notion of "finite splitting height" which generalises the finite-height property. By considering the dual CAT(0) cube complex associated to a codimension-1 subgroup H in G, we show that the Kropholler-Roller conjecture holds when H has finite splitting height in G. Examples of subgroups of finite height are stable subgroups or more generally strongly quasiconvex subgroups. Examples of subgroups of finite splitting height include relatively quasiconvex subgroups of relatively hyperbolic groups with virtually polycyclic peripheral subgroups. In particular, our results extend Stallings' theorem and generalise a theorem of Sageev on decomposing a hyperbolic group by quasiconvex subgroups.Comment: 11 pages, Corollary 1.6 has been added as an application of splittings over subgroups with property (T