ADDENDUM TO “LOWER ALGEBRAIC K-THEORY OF HYPERBOLIC 3-SIMPLEX REFLECTION GROUPS.”

CHARLES WEIBEL

Abstract. In this addendum we explicitly compute the Bass Nil-groups NK_{i}(ℤ[C_{4}]) for i = 0, 1 and NK_{0}(ℤ[D_{4}]). We also show that NK_{1}(ℤ[D_{4}]) is not trivial. Here C_{4} denotes the cyclic group of order 4 and D_{4} is the dihedral group of order 8.

In [LO], Lafont and Ortiz computed the lower algebraic K-theory of the integral group ring of all 32 hyperbolic 3-simplex reflection groups (see [LO, Tables 6–7]). For 25 of these integral group rings, their computation was completely explicit. For the remaining 7 examples, the expression for some of the K-groups involved the Bass Nil-groups NK_{0} and NK_{1} associated to D_{4} (the dihedral group of order 8).

In [L05], Lück computed the lower algebraic K-theory of the integral group ring of the semi-direct product of the three-dimensional discrete Heisenberg group by C_{4} (the cyclic group of order 4). These computations involved the Bass Nil-groups NK_{0} and NK_{1} associated to C_{4} (see [L05, Corollary 3.9]).

In this addendum we compute the Bass Nil-groups

NK_{n}(ℤG) = ker{K_{n}(ℤG[x]) ⟶^{x↦0} K_{n}(ℤG)},

where G is D_{2}, C_{4} or D_{4}, and n = 0, 1. We will use these calculations to complement the calculations of [L05] and [LO] in 1.5 and 2.9 below.

Our calculation will keep track of the additional structure on the groups NK_{n}(A) given by the Verschiebung and Frobenius operators, V_{m} and F_{m}, as well as the continuous module structure over the ring W (ℤ) of big Witt vectors; the additive group of W (ℤ) is the abelian group (1 + xℤ[[x]])^{×}. (See [We80] for more details.) In fact, it is a module over the slightly larger Cartier algebra which includes the Verschiebung and Frobenius operators V_{m} and F_{m} as well as the homotheties [a] for a ∈ ℤ; see 1 and 2.4 below.

It is convenient to write V for the continuous W (𝔽_{2})-module x𝔽_{2}[x], which, as an abelian group, is just a countable direct sum of copies of 𝔽_{2} = ℤ/2 on generators x^{i}, i > 0. The module structure on V is determined by V_{m}(x^{n}) = mn.

1. The groups C_{2}, D_{2} and C_{4}

For the cyclic group C_{2} = ⟨σ⟩ of order two, consider the Rim square:

(1)  \xymatrix{ℤ[C_{2}] \ar[d]_{σ↦−1} \ar[r]^{σ↦+1} & ℤ \ar[d]^{q} & or equivalently, & ℤ[C_{2}] \ar[d] \ar[r]^{σ↦(1,−1)} & ℤ × ℤ \ar[d]^{q×q} \\ ℤ \ar[r]^{q} & 𝔽_{2} &  & 𝔽_{2} \ar[r]^{Δ} & 𝔽_{2} × 𝔽_{2}}

from which we immediately get NK_{0}(ℤC_{2}) = NK_{1}(ℤC_{2}) = 0 as in [Bas68, XII.10.6] and [Mi71, 6.4]. From Guin-Loday-Keune [GWL80][Keu81], the double relative group NK_{2}(ℤC_{2}, σ +1, σ −1) is isomorphic to V, with the symbol ⟨x^{n}(σ −1), σ +1⟩ corresponding to x^{n} ∈ V. We also have a diagram:

\xymatrix{ & NK_{2}(ℤC_{2}, σ + 1, σ − 1) ≅ V \ar[d]^{≅} \\ 0 = NK_{3}(ℤ) \ar[r] & NK_{2}(ℤC_{2}, σ + 1) \ar[d] \ar[r]^{≅} & NK_{2}(ℤC_{2}) \ar[d] \ar[r] & NK_{2}(ℤ) = 0 \\  & 0 = NK_{2}(ℤ, 2) \ar[r] & NK_{2}(ℤ) = 0.}

Thus we obtain: Theorem 1.1. NK_{2}(ℤC_{2}) ≅ V with ⟨x^{n}(σ − 1), σ + 1⟩ corresponding to x^{n} ∈ V.

We now turn to the group D_{2} = C_{2} × C_{2}. First we need a calculation. Let Φ denote the Frobenius, so that Φ(V) = x^{2}𝔽_{2}[x^{2}], and write Ω_{R} for the Kähler differentials of R, so that Ω_{𝔽_{2}[x]} ≅ 𝔽_{2}[x]dx. Lemma 1.2. The map q : ℤ[C_{2}] → 𝔽_{2}[C_{2}] ≅ 𝔽_{2}[ε], ε^{2} = 0, in (1) induces an exact sequence

0 → Φ(V) → NK_{2}(ℤC_{2}) ⟶^{q} NK_{2}(𝔽_{2}[ε]) ⟶^{D} Ω_{𝔽_{2}[x]} → 0.

Proof. Van der Kallen computed NK_{2}(𝔽_{2}[ε]) in [vdK71, Ex. 3]: there is a split short exact sequence

(2)  0 → V/Φ(V) ⟶^{F} NK_{2}(𝔽_{2}[ε]) ⟶^{D} Ω_{𝔽_{2}[x]} → 0,

where F (x^{n}) = ⟨x^{n}ε, ε⟩ and D(⟨fε, g + g′ε⟩) = f dg. The map NK_{2}(ℤ[C_{2}]) ⟶^{q}

NK_{2}(𝔽_{2}[ε]) sends ⟨x^{n}(σ + 1), σ − 1⟩ to F (x^{n}) = ⟨x^{n}ε, ε⟩. By Theorem 1.1 and (2), this map has kernel Φ(V) and image F (V/Φ(V)). □

Remark. Although Ω_{𝔽_{2}[x]} is isomorphic to V as an abelian group, it has a different W (𝔽_{2})-module structure. This is determined by the formulas V_{m}(x^{n} dx) = mx^{mn−1} dx in Ω_{𝔽_{2}[x]}. Grunewald has pointed out that since Ω_{𝔽_{2}[x]} is not finitely generated as a module over the 𝔽_{2}-Cartier algebra (generated by the operators V_{m} and F_{m} and containing W (𝔽_{2})), neither are NK_{2}(𝔽_{2}[ε]) or (by 1.3 below) NK_{1}(ℤ[D_{2}]). Theorem 1.3. For D_{2} = C_{2} × C_{2}, NK_{0}(ℤ[D_{2}]) ≅ V, NK_{1}(ℤ[D_{2}]) ≅ Ω_{𝔽_{2}[x]} and the image of the map NK_{2}(ℤ[D_{2}]) → NK_{2}(ℤ[C_{2}])^{2} ≅ V^{2} is Φ(V) × V.

Proof. We tensor (1) with ℤC_{2}. Since 𝔽_{2}[C_{2}] ≅ 𝔽_{2}[ε], ε^{2} = 0, and NK_{1}(𝔽_{2}[C_{2}]) ≅ (1 + xε𝔽_{2}[x])^{×} ≅ V, then the Mayer-Vietoris sequence in [Mi71, Theorem 6.4] for the NK-functor,

\begin{matrix}NK_{2}(ℤ[D_{2}]) → (NK_{2}(ℤC_{2}))^{2} ⟶^{q×q} NK_{2}(𝔽_{2}[ε]) → NK_{1}(ℤ[D_{2}]) \\ → (NK_{1}(ℤ[C_{2}])^{2} → NK_{1}(𝔽_{2}[ε]) ⟶^{≅} NK_{0}(ℤ[D_{2}]) → NK_{0}(ℤC_{2}), \end{matrix} ; (3)

quickly gives NK_{0}(ℤ[D_{2}]) ≅ NK_{1}(𝔽_{2}[ε]) ≅ V. By Lemma 1.2, the initial portion of (3) yields the calculation of NK_{1}(ℤ[D_{2}]) and the asserted surjection NK_{2}(ℤ[D_{2}]) ↠ Φ(V) × V. □

ADDENDUM TO “LOWER ALGEBRAIC K-THEORY OF REFLECTION GROUPS.” 3

Remark. The kernel K of the map NK_{2}(ℤ[D_{2}]) → V^{2} in Theorem 1.3 has a subgroup generated by the double relative group NK_{2}(ℤ[D_{2}], σ_{1} + 1, σ_{1} − 1), which is isomorphic to 𝔽_{2}[ε] ⊗ V on the symbols ⟨x^{n}(a + bσ_{2})(σ_{1} + 1), σ_{2} − 1⟩, where σ_{1}, σ_{2} are the generators of D_{2} = C_{2} × C_{2}. The quotient of K by this subgroup is generated by the image of NK_{3}(𝔽_{2}[ε]), a group which I do not know.

The analysis for the cyclic group C_{4} of order 4 on generator σ is similar, using the Rim square

(4)  \xymatrix{ℤ[C_{4}] \ar[d]_{σ^{2}↦1} \ar[r]^{σ↦i} & ℤ[i] \ar[d]^{i↦1+ε} \\ ℤ[C_{2}] \ar[r]^{q} & 𝔽_{2}[ε]} .

Theorem 1.4. NK_{1}(ℤ[C_{4}]) ≅ Ω_{F_{2}[x]} and NK_{0}(ℤ[C_{4}]) ≅ V.

Proof. Since ℤ[i] is a regular ring, the Mayer-Vietoris sequence for (4) reduces to:

\begin{matrix}NK_{2}(ℤ[C_{4}]) ⟶^{0} NK_{2}(ℤC_{2}) ⟶^{q} NK_{2}(𝔽_{2}[ε]) → NK_{1}(ℤ[C_{4}]) \\ → NK_{1}(ℤ[C_{2}]) → NK_{1}(𝔽_{2}[ε]) ⟶^{≅} NK_{0}(ℤ[C_{4}]) → NK_{0}(ℤC_{2}). \end{matrix}

The isomorphism marked in this sequence follows from Theorem 1.1. By Lemma 1.2, the first map is zero and the cokernel of the map q is Ω_{𝔽_{2}[ε]}. □

Remark. As before, the group NK_{2}(ℤ[C_{4}]) has a subgroup generated by the double relative group NK_{2}(ℤ[C_{4}], σ^{2} + 1, σ^{2} − 1), isomorphic to 𝔽_{2}[ε] ⊗ V on ⟨σ^{2} + 1, x^{n}(σ^{2} − 1)⟩. The quotient is generated by the image of NK_{3}(𝔽_{2}[ε]), which I do not know.

Here is an application of this calculation. Let Hei denote the three-dimensional discrete Heisenberg group, which is the subgroup of GL(3, ℤ) consisting of upper triangular integral matrices with ones along the diagonal. Consider the action of the cyclic group C_{4} given by:

( \begin{matrix}1 & x & y \\ 0 & 1 & z \\ 0 & 0 & 1 \end{matrix} ) ↦ ( \begin{matrix}1 & −z & y − xz \\ 0 & 1 & x \\ 0 & 0 & 1 \end{matrix} ) .

Combining Theorem 1.4 with [L05, 3.9], the lower K-theory of the group Hei ⋊C_{4} is given in the following Corollary.

Proposition 1.5. We have: W h_{n}(Hei ⋊C_{4}) = \begin{cases}⨁_{∞} ℤ/2 & n = 0, 1; \\ 0 & n ≤ −1. \end{cases}

2. The dihedral group D_{4}

Before moving on to the group ring of D_{4}, we need some facts about the double relative groups K_{1}(A, B, I) when A → B is an injection. These groups were described in [GW83, 0.2] as:

(5) K_{1}(A, B, I) ≅ (B/A) ⊗ (I/I^{2})/{b ⊗ cz + c ⊗ zb − bc ⊗ z}, (b, c ∈ B, z ∈ I).

Moreover, by [GW83, 3.12 and 4.1], the map K_{1}(A, B, I) → K_{1}(A, I) sends the class of b ⊗ z to the class of the matrix ( \begin{matrix}1−zb & z \\ −bzb & 1+bz \end{matrix} ).

Lemma 2.1. Suppose that A → B is a ring homomorphism mapping an ideal I of A isomorphically onto an ideal of B. Then the double relative group satisfies

K_{1}(A[x], B[x], I[x]) ≅ K_{1}(A, B, I)⊗ℤ[x], and NK_{1}(A, B, I) ≅ K_{1}(A, B, I)⊗xℤ[x].

Proof. Because x is central in B[x], the formulas are immediate from (5). □

We will be specifically interested in the twisted group ring A = ℤ[i] ⋊ C_{2}, where C_{2} = ⟨τ⟩ acts on ℤ[i] by τiτ^{−1} = −i. It injects into the matrix ring B = M_{2}(ℤ) by the map ϕ : A → B defined by ϕ(i) = ( \begin{matrix}0 & 1 \\ −1 & 0 \end{matrix} ) and ϕ(τ) = ( \begin{matrix}1 & 0 \\ 0 & −1 \end{matrix} ). The ideal I = (2, 1 + τ)A maps isomorphically to 2B, and A/I ≅ 𝔽_{2}[ε_{1}], where ε_{1} = 1 + i and ε_{1}^{2} = 0. Hence we have the following cartesian square:

(6)  \xymatrix{A \ar[d]_{mod I} \ar[r]^{ϕ} & M_{2}(ℤ) \ar[d]^{mod 2B} \\ 𝔽_{2}[ε_{1}] \ar[r] & M_{2}(𝔽_{2}).}

To calculate NK_{1}(A), we use the following double relative calculation. Lemma 2.2. The double relative group K_{1}(A, B, 2B) of (6) is isomorphic to 𝔽_{2}, and NK_{1}(A, B, 2B) ≅ V. The map V ≅ NK_{1}(A, B, 2B) → NK_{1}(A) sends x^{n} ∈ V to the class of the unit 1 + x^{n}i(1 + τ) of A[x].

Proof. Since dim(B/A) = 2 and dim(I/I^{2}) = 4, the group (B/A) ⊗ (I/I^{2}) has 8 generators and 64 relations; a basis of B/A is {e_{11}, e_{12}} and the 2e_{ij} span I/I^{2}. By inspection of the relations in (5) we see that the map B/I ⊗ I/I^{2} → 𝔽_{2} sending e_{ij} ⊗2e_{kl} to δ_{il} +δ_{jk} sends A/I ⊗I/I^{2} and all the relations in (5) to zero, and sends e_{11} ⊗ 2e_{12} to 1. Thus it induces a surjection K_{1}(A, B, I) → 𝔽_{2}. We claim that this is an isomorphism.

The relations for (b, c, z) = (e_{11}, e_{11}, 2e_{11}), (e_{11}, e_{11}, 2e_{22}), (e_{11}, e_{12}, 2e_{12}) and (e_{11}, e_{21}, 2e_{21}) in (5) yields the relations:

0 = e_{11} ⊗ 2e_{11} = e_{11} ⊗ 2e_{22} = e_{12} ⊗ 2e_{12} = e_{21} ⊗ 2e_{21}.

The relations for (e_{12}, e_{11}, 2e_{11}), (b, c, z) = (e_{11}, e_{21}, 2e_{11}) and (e_{12}, e_{12}, 2e_{21}) in (5) yields the relations:

0 = e_{11} ⊗ 2e_{12} − e_{12} ⊗ 2e_{11} = e_{21} ⊗ 2e_{11} − e_{11} ⊗ 2e_{21} = e_{12} ⊗ 2e_{22} − e_{12} ⊗ 2e_{11}.

This verifies the claim, proving that K_{1}(A, B, 2B) ≅ 𝔽_{2}.

Finally, the map NK_{1}(A, B, 2B) → NK_{1}(A, I) sends the class of x^{n}e_{11} ⊗2e_{12} to

the class of the matrix ( \begin{matrix}1−2x^{n}e_{12}e_{11} & 2x^{n}e_{12} \\ −x^{2n}e_{11}e_{12}e_{11} & 1+2x^{n}e_{11}e_{12} \end{matrix} ) = ( \begin{matrix}1 & x^{n}i(1+τ) \\ 0 & 1+x^{n}i(1+τ) \end{matrix} ), which is the

class of 1 + x^{n}i(1 + τ) in NK_{1}(A). □

Remark. The elements u = i + τ and v = i(1 + τ) of A satisfy u^{2} = v^{2} = 0, and are distinct in A/2A = 𝔽_{2}[i, τ]. Hence the units (1+x^{m}u)(1+x^{n}v) of A[x] generate a subgroup of NK_{1}(A) isomorphic to V^{2}, which injects into NK_{1}(𝔽_{2}[i, τ]) ≅ V^{3}. Proposition 2.3. NK_{0}(A) = 0 and NK_{1}(A) ≅ V^{2} on the units (1+x^{m}u)(1+x^{n}v).

The map A → 𝔽_{2}[i, τ] ≅ 𝔽_{2}[ε_{1}, ε_{2}] and the map Ω_{𝔽_{2}[x]} ⟶^{δ} NK_{2}(𝔽_{2}[ε_{1}, ε_{2}]) sending

x^{n} to ⟨x^{n−1}ε_{1}ε_{2}, x⟩ induce a surjection NK_{2}(A) × Ω_{𝔽_{2}[x]} → NK_{2}(𝔽_{2}[ε_{1}, ε_{2}]).

Proof. Consider the Mayer-Vietoris sequence of the square (6). Since B = M_{2}(ℤ) and B/I = M_{2}(𝔽_{2}) are regular rings, NK_{∗}(B) = NK_{∗}(B/I) = 0 and hence NK_{∗}(B, I) = 0. We immediately get that NK_{∗}(A, B, I) ≅ NK_{∗}(A, I) and the Mayer-Vietoris sequence reduces to NK_{0}(A) ≅ NK_{0}(A/I) = 0, and the exact sequence:

NK_{2}(A) → NK_{2}(𝔽_{2}[ε_{1}]) → NK_{1}(A, B, I) → NK_{1}(A) → NK_{1}(𝔽_{2}[ε_{1}]) → 0.

By Lemma 2.2 and the remark preceding it, this yields the calculation of NK_{1}(A).

Now π : A → 𝔽_{2}[ε_{1}, ε_{2}] satisfies π(u) = ε_{1}+ε_{2}, π(v) = ε_{1}+ε_{1}ε_{2} and π(uv) = ε_{1}ε_{2}, so we may write 𝔽_{2}[ε_{1}, ε_{2}] ≅ 𝔽_{2}[ū, v̄]/(ū^{2}, v̄^{2}). By [vdK71], the group NK_{2}(𝔽_{2}[ū, v̄]) is isomorphic to the direct sum of NK_{2}(𝔽_{2}[ū]), NK_{2}(𝔽_{2}[v̄]) and a group with the following generators:

⟨x^{n}ū, v̄⟩, ⟨x^{n}ūv̄, ū⟩, ⟨x^{n}ūv̄, v̄⟩ and ⟨x^{n−1}ūv̄, x⟩.

Since u^{2} = v^{2} = 0 in A, all these symbols lift to Dennis-Stein symbols in NK_{2}(A) except possibly the symbols ⟨x^{n−1}ūv̄, x⟩. But these symbols are hit by the image of Ω_{𝔽_{2}[x]} under δ. □

Remark. δ : Ω_{𝔽_{2}[x]} → NK_{2}(𝔽_{2}[ε_{1}, ε_{2}]) is a homomorphism by the Dennis-Stein identity ⟨f, x⟩⟨g, x⟩ = ⟨f + g − fgx, x⟩ with fg = 0; see [GWL80, p. 184]. It is a morphism of Cartier modules since V_{m}⟨x^{n−1}ε_{1}ε_{2}, x⟩ = m⟨x^{mn−1}ε_{1}ε_{2}, x⟩ = δ(V_{m}(x^{n})).

Our analysis of D_{4} will involve the units of the ring ℤ/4[x][C_{2}].

Example 2.4. Consider the modular group ring B = ℤ/4[C_{2}] = ℤ/4[e]/(e^{2} − 2e), with e = 1−τ. The ideals 2eB of B and eB/2eB of B/2eB are isomorphic to 𝔽_{2}, so both NK_{1}(B, 2e) and NK_{1}(B/2e, e) are isomorphic to V and the group NK_{1}(B, e), identified with the abelian group (1 + xeB[x])^{×} is a nontrivial extension:

0 → V → NK_{1}(B, e) → V → 0.

As an abelian group, NK_{1}(B, e) is the direct sum of a countably infinite free ℤ/4module on the (1 + ex^{m}) (m = 1, 2, …) and a countably infinite free ℤ/2-module on the (1 + 2ex^{2i−1}) (i = 1, 2, …). As a module over the ℤ/4-Cartier algebra (generated by the operators V_{m}, F_{m} and homothety [2]), NK_{1}(B, e) is cyclic on generator u = 1 + ex; V_{m}(u) = 1 + ex^{m} and V_{m}[2](u) = 1 + 2ex^{m}.

Finally, we are in position to analyze NK_{0}(ℤ[D_{4}]). The sharp exponent 4 for NK_{0}(ℤ[D_{4}]) in Theorem 2.5 is a slight improvement on the bound in [CP02]. It is convenient to write D_{4} as the semidirect product of C_{4} (on σ) with the cyclic group C_{2} = {1, τ}, with relation τστ = σ^{−1}.

Theorem 2.5. The group NK_{0}(ℤ[D_{4}]) is isomorphic to the cyclic Cartier module NK_{1}(ℤ/4[C_{2}], 1 − τ), described in Example 2.4. As a group, it is the direct sum of a countably infinite free ℤ/4-module and a countably infinite free ℤ/2-module.

Proof. We can map ℤ[D_{4}] to the twisted ring A = ℤ[i] ⋊ C_{2} occurring in (6) above, sending σ to i. Combining this with the natural surjection onto the subring ℤ[D_{2}] of ℤ[C_{2}] × ℤ[C_{2}], we get a ring map ℤ[D_{4}] → A × ℤ[C_{2}] × ℤ[C_{2}]. The ideal I = (4, 2−2σ, σ^{2}−1)ℤ[D_{4}] has B_{0} = ℤ[D_{4}]/I = ℤ/4[D_{2}]/(2−2σ), and is isomorphic to the ideal 2A × (4) × (4) of A × ℤ[C_{2}] × ℤ[C_{2}]. Consider the following cartesian square:

(7)  [figure: ℤ[D_{4}]; σ↦(i,1,−1); τ↦(τ,τ,τ); A × ℤ[C_{2}] × ℤ[C_{2}]; π; B_{0} = ℤ/4[D_{2}]/(2 − 2σ); q=(q_{0},q_{+},q_{−}); 𝔽_{2}[D_{2}] × B × B.]

The kernel of the split surjection q_{+} : B_{0} → B = ℤ/4[C_{2}] is the 2-dimensional ideal J = (1 − σ)B_{0}. This implies that NK_{1}(B_{0}) = NK_{1}(B) ⊕ NK_{1}(B_{0}, J). Because NK_{1}(ℤ[C_{2}]) = NK_{0}(ℤ[C_{2}]) = NK_{0}(A) = 0 (by 2.3), the Mayer-Vietoris sequence associated to (7) ends

(8) NK_{1}(A) × NK_{1}(B_{0}, J) ⟶^{η} NK_{1}(𝔽_{2}[D_{2}]) × NK_{1}(B) → NK_{0}(ℤ[D_{4}]) → 0,

The displayed map η is given by the matrix (\begin{matrix}π & 0 \\ q_{0} & q_{−} \end{matrix}) . It is easy to see that NK_{1}(B_{0}, J) is isomorphic to V^{2} on the terms (1+(1−σ)x^{m}) and (1+(1−σ)τx^{n}). An elementary calculation using the isomorphism NK_{1}(A) ≅ V^{2} of 2.3 shows that η is an injection, sending the module NK_{1}(A)×NK_{1}(B_{0}, J) ≅ V^{4} isomorphically onto the subgroup NK_{1}(𝔽_{2}[D_{2}]) × NK_{1}(ℤ/4). Since NK_{1}(B) = NK_{1}(ℤ/4) ⊕ NK_{1}(B, eB), e = 1 − τ, it follows that the induced map NK_{1}(B, e) → NK_{0}(ℤ[D_{4}]) is an isomorphism. □

To begin the calculation of NK_{1}(ℤ[D_{4}]), we extend the Mayer-Vietoris sequence (8) associated to (7) to the left. This is possible by the following observation: since B_{0} maps onto each of the three ring factors on the lower right of (7), the presentation (5) shows that the double relative K_{1} obstruction vanishes. Because η is an injection in (8), the continuation of the Mayer-Vietoris sequence yields the exact sequence:

(9)  (∗) \underrightarrow{(\begin{matrix}π & 0 \\ q_{0} & q_{−} \end{matrix})} NK_{2}(𝔽_{2}[D_{2}]) × NK_{2}(B) → NK_{1}(ℤ[D_{4}]) → 0.

where (∗) denotes NK_{2}(A) × NK_{2}(ℤ[C_{2}])^{2} × NK_{2}(B_{0}, J). Definition 2.6. The map ∪[x] : NK_{0}(ℤ[D_{4}]) → NK_{1}(ℤ[D_{4}]) is obtained by composing the isomorphism NK_{1}(B) ≅ NK_{0}(ℤ[D_{4}]) of Theorem 2.5 with the canonical map NK_{2}(B, e) → NK_{2}(B) → NK_{1}(ℤ[D_{4}]) of (9).

There is a canonical map NK_{1}(B, e) → NK_{2}(B, e) sending the unit 1 − aex to ⟨ae, x⟩; the composition with NK_{2}(B, e) ⊂ K_{2}(B[x, x^{−1}], e) is given by 1 − aex ↦ {1 − aex, x} (multiplication by the class of x in K_{1}(ℤ[x, x^{−1}])).

The analogous maps from V ≅ NK_{1}(B, 2e) and V ≅ NK_{1}(B/2eB, e) to Ω_{𝔽_{2}[x]} ≅ NK_{2}(B, 2e) and Ω_{𝔽_{2}[x]} ≅ NK_{2}(B/2eB, e) are compatible with the divided power map [d] : V → Ω_{𝔽_{2}[x]} sending x^{n} to x^{n−1} dx. Note that [d] is an isomorphism of abelian groups but is not a morphism of 𝔽_{2}-Cartier modules. Theorem 2.7. The map ∪[x] : NK_{0}(ℤ[D_{4}]) → NK_{1}(ℤ[D_{4}]) in 2.6 is a surjection. Hence the group NK_{1}(ℤ[D_{4}]) has exponent 2 or 4, and there is a commutative diagram whose rows are exact:

\xymatrix{0 \ar[r] & V \ar[d]^{[d]}_{≅} \ar[r] & NK_{0}(ℤ[D_{4}]) \ar[d]^{∪[x]} \ar[r] & V \ar[d]^{[d]}_{≅} \ar[r] & 0 \\  & Ω_{𝔽_{2}[x]} \ar[r] & NK_{1}(ℤ[D_{4}]) \ar[r] & Ω_{𝔽_{2}[x]} \ar[r] & 0.}

Proof. A diagram chase on (9) shows that NK_{1}(ℤ[D_{4}]) is an extension of the cokernel of NK_{2}(ℤ[C_{2}]) × NK_{2}(B_{0}, J) → NK_{2}(B) by a quotient of the cokernel of NK_{2}(A) → NK_{2}(𝔽_{2}[D_{2}]). These cokernels are both Ω_{𝔽_{2}[x]}, by Proposition 2.3 and Lemma 2.8 below, yielding the bottom row of the theorem. The map ∪[x] sends the element corresponding to 1 − x^{n}ae ∈ NK_{1}(B, e) to the element corresponding to ⟨x^{n−1}ae, x⟩ ∈ NK_{2}(B, e), so the diagram in the theorem commutes by inspection. □

Lemma 2.8. The cokernel of the map NK_{2}(ℤ[C_{2}]) × NK_{2}(B_{0}, J) → NK_{2}(B) in (9) is Ω_{𝔽_{2}[x]}, on symbols ⟨x^{n−1}e, x⟩.

Proof. The kernel of the map q_{−} : B_{0} → B is the ideal J′ = (1 + σ)B_{0}. Because J ∩J′ = 0 in B_{0}, the double relative group NK_{2}(B_{0}, J, J′) is isomorphic to 𝔽_{2}[C_{2}][x] on symbols ⟨x^{m}(1+σ), (1−σ)⟩ and ⟨x^{m}τ(1+σ), (1−σ)⟩ by [GWL80][Keu81]. Since J ≅ 2B, we have an exact sequence

(10)  𝔽_{2}[C_{2}][x] → NK_{2}(B_{0}, J) ⟶^{q_{−}} NK_{2}(B, 2B) → 0.

Combining this with the ideal sequence for 2B ⊂ B shows that the cokernel of NK_{2}(B_{0}, J) → NK_{2}(B) is NK_{2}(B/2B). Since B/2B ≅ 𝔽_{2}[C_{2}], the lemma now follows from Lemma 1.2. □

Inserting the calculations of Theorems 2.5 and 2.7 into Tables 6–7 in [LO], we obtain the following result.

Theorem 2.9. Let Γ be one of the following hyperbolic 3-simplex reflection groups: [(3, 4, 3, 6)], [4, 3^{[3]}], [4, 3, 6], [(3^{3}, 4)], [4, 3, 5], [(3, 4)^{[2]}], [(3, 4, 3, 5)]. Then the lower algebraic K-theory of the groups Γ is given by the following table:

\begin{matrix}Γ & K_{−1} ≠ 0 & K̃_{0} ≠ 0 & W h ≠ 0 \\ [(3, 4, 3, 6)] & ℤ^{3} & (ℤ/4)^{2} ⊕ Nil_{0} & Nil_{1} \\ [4, 3^{[3]}] & ℤ^{3} & (ℤ/4)^{2} ⊕ Nil_{0} ⊕ ⨁_{∞} ℤ/2 & Nil_{1} ⊕ ⨁_{∞} ℤ/2 \\ [4, 3, 6] & ℤ^{4} & (ℤ/4)^{2} ⊕ Nil_{0} ⊕ ⨁_{∞} ℤ/2 & Nil_{1} ⊕ ⨁_{∞} ℤ/2 \\ [(3^{3}, 4)] & ℤ^{2} & (ℤ/4)^{2} ⊕ Nil_{0} ⊕ ⨁_{∞} ℤ/2 & Nil_{1} ⊕ ⨁_{∞} ℤ/2 \\ [4, 3, 5] & ℤ^{3} & (ℤ/4)^{2} ⊕ Nil_{0} ⊕ ⨁_{∞} ℤ/2 & ℤ^{3} ⊕Nil_{1} ⊕ ⨁_{∞} ℤ/2 \\ [(3, 4)^{[2]}] & ℤ^{4} & (ℤ/4)^{4} ⊕ Nil_{0} ⊕ ⨁_{∞} ℤ/2 & Nil_{1} ⊕ ⨁_{∞} ℤ/2 \\ [(3, 4, 3, 5)] & ℤ^{4} & (ℤ/4)^{2} ⊕ Nil_{0} ⊕ ⨁_{∞} ℤ/2 & ℤ^{3} ⊕Nil_{1} ⊕ ⨁_{∞} ℤ/2 \end{matrix}

In this table, Nil_{0} = NK_{i}(ℤ[D_{4}]) is the direct sum of a countably infinite free ℤ/4-module and a countably infinite free ℤ/2-module, and Nil_{1} = NK_{1}(ℤ[D_{4}]) is a countably infinite torsion group of exponent 2 or 4.

References

[Bas68] H. Bass, Algebraic K-theory, W. A. Benjamin, New York, 1968.

[CP02] F. Connolly and S. Parasidis, On the exponent of the cokernel of the forget-control map on K_{0}-groups, Fund. Math. 172 (2002), 201–216.

[F87] F. T. Farrell, A remark on K_{0} of crystallographic groups, Topology Appl. 26 (1987), 97–99.

[GW83] S. Geller and C. Weibel, K_{1}(A, B, I), J. reine angew. Math. 342 (1983), 12–34.

[GWL80] D. Guin-Waléry, and J-L. Loday, Obstruction à l’excision en K-théorie algébrique, Algebraic K-theory, Evanston 1980 (Proc. Conf. Northwestern Univ., Evanston, Ill., 1980), pp. 179–216, Lecture Notes in Math., 854, Springer, Berlin, 1981.

[Keu81] F. Keune, Doubly relative K-theory and the relative K_{3}, J. Pure Appl. Algebra 20 (1981), 39–53.

[LO] J. F. Lafont, and I. J. Ortiz, Lower algebraic K-theory of hyperbolic 3-simplex reflection groups, to appear in Comment. Math. Helv.

[L05] W. Lück, K- and L-theory of the semi-direct product of the discrete 3-dimensional Heisenberg group by ℤ/4. Geom. Topol. 9 (2005), 1639–1676

[Mi71] J. Milnor, Introduction to algebraic K-theory, Princeton Univ. Press, Princeton, 1971.

[vdK71] W. van der Kallen, Le K_{2} des nombres duaux, C. R. Acad. Sci. Paris Sér. A-B 273 (1971), 1204–1207.

[We80] Mayer-Vietoris sequences and module structures on NK_{∗}, pp. 466–493 in Lecture Notes in Math., #854, Springer-Verlag, 1981.

Department of Mathematics, Rutgers University, Piscataway, NJ 08854 E-mail address, Charles Weibel: weibel@math.rutgers.edu