## Algebraic $$K$$-theory over the infinite dihedral group: an algebraic approach.(English)Zbl 1236.19002

The infinite dihedral groups has two descriptions. The first is as the free product of two cyclic groups of order $$2$$. The second is as the non-trivial semidirect product of the infinite cyclic group by the cyclic group of order $$2$$. The second description implies that the infinite dihedral group has an infinite cyclic subgroup of index $$2$$.
So if a group $$G$$ admits an epimorphism to the infinite dihedral group, first of all, it admits a splitting as an amalgamated free product. Furthermore, it contains a subgroup $$\bar{G}$$ of index $$2$$, that is a semidirect product of a group by the infinite cyclic group. One direct summand of the algebraic $$K$$-theory of $$RG$$ ($$R$$ a ring) is the Waldhausen’s Nil-groups, the other is its controlled part over the interval. There is a similar splitting for the $$K$$-theory of $$R\bar{G}$$. In this case the Nil-groups split also into two summands, each summand being a Bass-Farrell-Hsiang Nil-group.
In this paper, the authors prove that the Nil-groups for $$RG$$ are isomorphic to one of the Nil-groups of $$R\bar{G}$$ (the Nil-Nil Theorem). The proof is a direct calculation on the Nil-categories and their $$K$$-theory. The first very interesting application of the result is an improvement of the Farrell-Jones Isomorphism Conjecture for $$K$$-theory. The original conjecture predicted that the $$K$$-theory of a group $$G$$ can be calculated from the $$K$$-theory of its virtually cyclic subgroups. Using, their results the authors prove that we need to show that the $$K$$-theory of a group can be calculated from their finite-by-cyclic subgroups. As an application of their more refined version of the conjecture, the authors compute the $$K$$-theory of the modular group.
The Nil-groups appear as obstructions to splitting problems in Topology. The authors define a map $$f: M \to X$$ to be semisplit along $$N \subset M$$, that separates $$M$$, if the map induces an isomorphism on fundamental groups, and it is simple homotopic to a map $$h: (M, N) \to (X, Y)$$ is, essentially, a homotopy equivalence of the one side of the splitting i.e., if $$M - N = M_1\cup M_2$$, $$X - Y = X_1\cup X_2$$, then $$h$$ induces a homotopy equivalence between the pairs $$(M_2, N)$$ and $$(X_2, Y)$$. Their main result is that if $${\pi}(Y)$$ has finite index in $${\pi}_(M_2)$$, then $$f$$ is semisplit.
The results of this paper are the algebraic approach to similar results in [J. F. Davis, F. Quinn and H. Reich, J. Topol. 4, No. 3, 505–528 (2011; Zbl 1227.19004)].

### MSC:

 19D35 Negative $$K$$-theory, NK and Nil 57R19 Algebraic topology on manifolds and differential topology

### Keywords:

Nil group; $$K$$-theory; Farrell-Jones conjecture

Zbl 1227.19004
Full Text:

### References:

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