On the Farrell-Jones Isomorphism Conjecture
Kun Wang · OhioLink ETD Center (Ohio Library and Information Network) · 2014
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. Lück 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 I would like to thank: My advisor Jean Lafont for his guidance and continuous support throughout the years.The thesis would not have been possible without the advising of Jean.Mike Davis and his wife Wanda Davis for their support and kindness.It was always very pleasant to talk to them.