Ivonne Johanna Ortiz

Lower algebraic K-theory of hyperbolic 3-simplex reflection groups

A hyperbolic 3-simplex reflection group is a Coxeter group arising as a lattice in O C.3;1/, with fundamental domain a geodesic simplex in H 3 (possibly with some ideal ver- tices). The classification of these groups is known, and there are exactly 9 cocompact examples, and 23 non-cocompact examples. We provide a complete computation of the lower algebraic K-theory of the integral group ring of all the hyperbolic 3-simplex reflection groups. Mathematics Subject Classification (2000). 19A31, 19B28, 19D35, 18F25, 16E20.

Relating the Farrell Nil-groups to the Waldhausen Nil-groups

Abstract Every virtually cyclic group Γ that surjects onto the infinite dihedral group D ∞ contains an index two subgroup ∏ of the form We show that the Waldhausen Nil-group of Γ vanishes if and only if the Farrell Nil-group of ∏ vanishes. 2000 Mathematics Subject Classification: 19D35.

Lower algebraic K-theory of certain reflection groups

For P ⊂ 3 a finite volume geodesic polyhedron, with the property that all interior angles between incident faces are of the form π/ m ij ( m ij ≥ 2 an integer), there is a naturally associated Coxeter group Γ P . Furthermore, this Coxeter group is a lattice inside the semi-simple Lie group O + (3, 1) = Isom( 3 ), with fundamental domain the original polyhedron P . In this paper, we provide a procedure for computing the lower algebraic K -theory of the integral group ring of such groups Γ P in terms of the geometry of the polyhedron P . As an ingredient in the computation, we explicitly calculate the K −1 and Wh of the groups D n and D n × 3 , and we also summarize what is known about the 0 .

Splitting formulas for certain Waldhausen Nil-groups

We provide splitting formulas for certain Waldhausen Nil-groups. We focus on Waldhausen Nilgroups associated to acylindrical amalgamations Γ = G1 ∗H G2 of the groups G1 and G2 over a common subgroup H. For these amalgamations, we explain how, provided that G1,G2 and Γ satisfy the Farrell–Jones isomorphism conjecture, the Waldhausen Nil-groups Nil W (RH;R[G1 − H],R[G2 − H]) can be expressed as a direct sum of Nil-groups associated to a specific collection of virtually cyclic subgroups of Γ. A special case covered by our theorem is the case of arbitrary amalgamations over a finite group H. Waldhausen’s Nil-groups were introduced in the two seminal papers [35, 36]. The motivation behind these Nil-groups originated from a desire to have a Mayer–Vietoris type sequence in algebraic K-theory. More precisely, if a group Γ = G1 ∗H G2 splits as an amalgamation of two groups G1 and G2 over a common subgroup H, then one can ask how the algebraic K-theory of the group ring R Γ is related to the algebraic K-theory of the integral group rings RG1, RG2 ,a ndRH. Motivated by the corresponding question in homology (or cohomology), one might expect a Mayer–Vietoris type exact sequence:

Equivariant K-homology for hyperbolic reflection groups

We compute the equivariant K-homology of the classifying space for proper actions, for cocompact 3-dimensional hyperbolic reflection groups. This coincides with the topological K-theory of the reduced C∗-algebra associated to the group, via the Baum-Connes conjecture. We show that, for any such reflection group, the associated K-theory groups are torsion-free. As a result we can promote previous rational computations to integral computations. Our proof relies on a new efficient algebraic criterion for checking torsion-freeness of K-theory groups, which could be applied to many other classes of groups.

Fundamental Domains for Actions on Spaces of Planes

Theorem 5.1 showed that the lower algebraic K-theory of any crystallographic group can be computed in two pieces. In Chap. 7, we completed the first half of the computation for the 73 split three-dimensional crystallographic groups. The results obtained are summarized in Table 7.8.

Fundamental Domains for the Maximal Groups

The classification of split three-dimensional crystallographic groups from Theorem 4.2 shows that seven of the groups contain all of the others as subgroups. For i = 1, …, 7, we let \(\varGamma _{i} =\langle L_{i},H_{i}\rangle\), where L i is the ith lattice (in the order that the, where L i is the ith lattice (in the order that the lattices are listed in Table 4.1) and H i is the maximal point group to be paired with L i . For instance,

Arithmetic Classification of Pairs (L, H)

\displaystyle{\varGamma _{4} = \left \langle \left \langle \frac{1} {2}(\mathbf{x} + \mathbf{z}),\mathbf{y},\mathbf{z}\right \rangle,D_{2}^{+} \times (-1)\right \rangle.}