Computations of K- and L-Theory of Cocompact Planar Groups

Abstract

Using the isomorphism conjectures of Baum & Connes and Farrel & Jones, we compute the algebraic K-and L-theory and the topological K-theory of cocompact planar groups (= cocompact N.E.C-groups) and of groups G appearing in an extension 1 → Z n → G → π → 1 where π is a finite group and the conjugation π-action on Z n is free outside 0 ∈ Z n . These computations apply for instance to two-dimensional crystallographic groups and cocompact Fuchsian groups.

The Novikov Conjecture for Linear Groups

“…Indeed if BΓ is the quotient of EΓ by Γ then there is a map from K Γ n (EΓ ) to K n (BΓ ) for which the composition K n (BΓ ) → K Γ n (EΓ ) → K n (BΓ ) is induced by the natural map from BΓ to BΓ . Standard arguments show that the induced map is an isomorphism after inverting the primes in R (compare Lemma 2.8 in [23]). Putting together (7.5) and (7.6) we obtain the split injection (7.2).…”

A Caveat on the Isomorphism Conjecture in L-theory

“…Then the Farrell-Jones-Conjecture with respect to the family VC holds for L for all decorations as explained in [6,Remark 4.32, Remark 6.5]. This fits together with the fact that in the first case B = VC ∪ SU B(Z n ) and in the second case B = VC.…”

“…We will follow the notation and setup of [6], which is different, but equivalent to the original setup in [5], [8], and slightly more convenient for our purposes. Let G be a group.…”

The lower algebraic K-theory of Γ3

“…Using group cohomology (see [Bro82, Theorem IV.3.12, Theorem IV.6.6]) a fairly straight-forward calculation shows that Q must be one of the following groups (similar calculations also appear for example in [BFPP00], [LS00], [U96]):…”