AbstractsMathematics

In this thesis, we study three different aspects of the Farrell-Jones Conjecture (FJC). The first is the study of the conjecture for groups admitting nice but not necessarily proper actions on CAT(0)-spaces (stabilizers can be infinite). It is a natural question that if the point stabilizers of the action satisfy the conjecture, whether the original group satisfies the conjecture. For this, we introduce the notion of hyperdiscrete group actions. Every proper action is hyperdiscrete. There are many other interesting examples. It turns out this new notion of group actions mostly fit into the framework for proving FJC developed by A. Bartels, W. Lueck and H. Reich. The second is the study of inheritance properties of the conjecture. We study the problem that if a group has a subgroup of finite index satisfying the conjecture, whether the group itself satisfies the conjecture. We reduce the problem to a special case and results obtained for this special case strongly suggests the rationalized conjecture is invariant under commensuration. The third part of this thesis is a joint work with J. Lafont and S. Prassidis. We study the Farrell Nil-groups associated to a virtually cyclic group, which is the obstruction to reduce the family of virtually cyclic groups used in FJC to the family of finite groups. We indeed study the more general Farrell Nil-groups associated to a finite order automorphism of a ring R. We show that any such Farrell Nil-group is either trivial, or infinitely generated (as an abelian group). Building on this first result, we then show that any finite group that occurs in such a Farrell Nil-group occurs with infinite multiplicity. If the original finite group is a direct summand, then the countably infinite sum of the finite subgroup also appears as a direct summand. We use this to deduce a structure theorem for countable Farrell Nil-groups with finite exponent. Finally, as an application, we show that if V is any virtually cyclic group, then the associated Farrell or Waldhausen Nil-groups can always be expressed as a countably infinite sum of copies of a finite group, provided they have finite exponent (which is always the case in dimension 0).