We discuss an analogon to the Farrell-Jones Conjecture for homotopy algebraic K-theory. In particular, we prove that if a group G acts on a tree and all isotropy groups satisfy this conjecture, then G satisfies this conjecture. This result can be used to get rational injectivity results for the assembly map in the Farrell-Jones Conjecture in algebraic K-theory. Key words: K-theory and homotopy K-theory of group rings, Isomorphism Conjectures, Actions on trees.

The Farrell-Jones and the Baum-Connes Conjecture say that one can compute the algebraic K- and L-theory of the group ring and the topological K-theory of the reduced group C ∗-algebra of a group G in terms of these functors for the virtually cyclic subgroups or the finite subgroups of G. By induction theory we want to reduce these families of subgroups to a smaller family, for instance to the family of subgroups which are either finite hyperelementary or extensions of finite hyperelementary groups with Z as kernel or to the family of finite cyclic subgroups. Roughly speaking, we extend the induction theorems of Dress for finite groups to infinite groups. Key words: K- and L-groups of group rings and group C ∗-algebras, induction theorems.

Abstract. It is proved that the assembly maps in algebraic K- and L-theory with respect to the family of finite subgroups is injective for groups Γ with finite asymptotic dimension that admit a finite model for EΓ. The result also applies to certain groups that admit only a finite dimensional model for EΓ. In particular, it applies to discrete subgroups of virtually connected Lie groups.

Buildings were first introduced by J.Tits in 1950s to give systematic geometric interpretations of exceptional Lie groups and have been generalized in various ways: Euclidean buildings (Bruhat-Tits buildings), topological buildings, R-buildings, in particular R-trees. They are useful for many different applications in various subjects: algebraic groups, finite groups, finite geometry, representation theory over local fields, algebraic geometry, Arakelov intersection for arithmetic varieties, algebraic K-theories, combinatorial group theory, global geometry and algebraic topology, in particular cohomology groups, of arithmetic groups and S-arithmetic groups, rigidity of cofinite subgroups of semisimple Lie groups and nonpositively curved manifolds, classification of isoparametric submanifolds in R n of high codimension, existence of hyperbolic structures on three dimensional manifolds in Thurston’s geometrization program. In this paper, we survey several applications of buildings in differential geometry and geometric topology. There are four underlying themes in these applications: 1. Buildings often describe the geometry at infinity of symmetric spaces and locally symmetric

Abstract. In this work, the assembly map in L-theory for the family of finite subgroups is proven to be a split injection for a class of groups. Groups in this class, including virtually polycyclic groups, have universal spaces that satisfy certain geometric conditions. The proof follows the method developed by Carlsson-Pedersen to split the assembly map in the case of torsion free groups. Here, the continuously controlled techniques and results are extended to handle groups with torsion. 1.

Abstract. In this paper we show that the Farrell-Jones isomorphism conjectures are inherited in group extensions for assembly maps in K-theory and L-theory with twisted coefficients.

Abstract. In this work, the continuously controlled techniques developed by Carlsson and Pedersen are used to prove that the Baum-Connes map is a split injection for groups satisfying certain geometric conditions. 1.

On the domain of the assembly map in algebraic K–theory