**Rational equivariant \(K\)-homology of low dimensional groups. **(joint with J.-F. Lafont and R. Sánchez-García),

33 pages as a preprint. Final version in Clay Math. Proceedings 16 (2012), pgs. 131-164.

The volume is entitled Topics in Noncommutative Geometry, Proceedings of the 3rd Winter School at the Luis Santaló-CIMPA Research School (Buenos Aires, 2010).

We consider groups \(G\) which have a cocompact 3-manifold model for the classifying space for proper \(G\)-actions. We provide an algorithm for computing the rationalized equivariant \(K\)-homology of the classifying space. Under the additional hypothesis that the \(G\)-action on the 3-dimensional model is smooth, the Baum-Connes conjecture holds, and the rationalized \(K\)-homology groups of the classifying space coincide with the rationalized topological \(K\)-theory of the reduced \(C^*\)-algebra of \(G\). We illustrate our algorithm on several concrete examples.

**Lower Algebraic \(K\)-theory of hyperbolic reflection group.** (joint with B. Magurn and J.-F. Lafont).

35 pages as a preprint, PDF. Final version in Math. Proc. Cambridge Philos. Soc. 148 (2010), pgs. 193-226.

A 3-dimensional hyperbolic reflection group is a Coxeter group arising as a lattice in the isometry group of hyperbolic 3-space, with fundamental domain a geodesic finite volume geodesic polyhedron \(P\). Building on our previous work (the case where \(P\) was a tetrahedron), we provide formulas for the loweralgebraic \(K\)-theory of the integral group ring of all the 3-dimensional hyperbolic reflection groups. The expressions for the lower algebraic \(K\)-theory end up depending solely on the internal dihedral angles between the faces of the polyhedron \(P\). In particular, the computation of the lower algebraic \(K\)-theory of such groups reduces to the computation of the lower algebraic \(K\)-theory of dihedral groups, as well as products of dihedral groups with the cyclic group of order two.

**Lower Algebraic \(K\)-theory of hyperbolic 3-simplex reflection groups.** (joint with J.-F. Lafont).

PDF. 33 pages as a preprin. Final version in Comment. Math. Helv. 84 (2009), pgs.297-337.

A 3-dimensional hyperbolic reflection group is a Coxeter group arising as a lattice in the isometry group of hyperbolic 3-space, with fundamental domain a geodesic finite volume geodesic polyhedron P. Building on our previous work (the case where P was a tetrahedron), we provide formulas for the lower algebraic \(K\)-theory of the integral group ring of all the 3-dimensional hyperbolic reflection groups. The expressions for the lower algebraic \(K\)-theory end up depending solely on the internal dihedral angles between the faces of the polyhedron P. In particular, the computation of the lower algebraic \(K\)-theory of such groups reduces to the computation of the lower algebraic \(K\)-theory of dihedral groups, as well as products of dihedral groups with the cyclic group of order two.

**Splitting formulas for certain Waldhausen Nil-groups.** (joint with J.-F. Lafont).

PDF. 16 pages as a preprint. Final version in J. London Math. Soc. 79 (2009), pgs.309-322.

For a group \(G\) that splits as an amalgamation of \(A\) and \(B\) over a common subgroup \(C\), there is an associated Waldhausen Nil-group, measuring the "failure" of Mayer-Vietoris for algebraic \(K\)-theory. Assume that (1) the amalgamation is acylindrical, and (2) the groups \(A\),\(B\),\(G\) satisfy the Farrell-Jones isomorphism conjecture. Then we show that the Waldhausen Nil-group splits as a direct sum of Nil-groups associated to certain (explicitly describable) infinite virtually cyclic subgroups of \(G\). We note that a special case of an acylindrical amalgamation includes any amalgamation over a finite group. Taken in combination with recent work by several mathematicians (J. Davis, Q. Khan, A. Ranicki, H. Reich, and F. Quinn), this completely reduces (modulo the Farrell-Jones isomorphism conjecture) the computation of Waldhausen Nil-groups associated to acylindrical amalgamations to the considerably easier computation of Farrell Nil-groups associated with various virtually cyclic subgroups.

**Relating the Farrell Nil-groups to the Waldhausen Nil-groups.** (joint with J.-F. Lafont).

PDF. 10 pages as a preprint. Final version in Forum Math. 20 (2008), pgs.445-455.

We prove that the Waldhausen Nil-groups associated to a virtually cyclic group that surjects onto the infinite dihedral group vanishes if and only if the Farrell Nil-group associated to the canonical index two subgroup is trivial. The proof uses the transfer map to establish one direction, and uses controlled pseudo-isotopy techniques of Farrell-Jones to establish the reverse implication.

**Relative hyperbolicity, classifying spaces, and lower algebraic \(K\)-theory. ** (joint with J.-F. Lafont).

PDF. 28 pages as a preprint. Final version in Topology 46 (2007), pgs. 527-533.

For \(G\) a relatively hyperbolic group, we provide a recipe for constructing a model for the universal space among G-spaces with isotropy in the family of virtually cyclic subgroups of \(G\). For \(G\) a Coxeter group acting as a non-uniform lattice on hyperbolic 3-space, we construct the classifying space explicitly, resulting in an 8-dimensional classifying space. We use the classifying space we obtain to compute the lower algebraic \(K\)-theory for one of these Coxeter groups.

**Erratum: The Lower Algebraic \(K\)-theory of \(\Gamma_3\). **

PDF. 2 pages as a preprint. Final version in \(K\)-theory 38 (2007), pgs. 85-86

The lower algebraic \(K\)-theory of \(\Gamma_3\) presented in the Main Theorem of the paper listed below is incorrect. In this erratum we present the correct version of this theorem.