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Register a new userLower algebraic K-theory of certain reflection groups
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For a finite volume geodesic polyhedron P in hyperbolic 3-space, with the property that all interior angles between incident faces are integral submultiples of Pi, there is a naturally associated Coxeter group generated by reflections in the faces. Furthermore, this Coxeter group is a lattice inside the isometry group of hyperbolic 3-space, 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 Coxeter lattices in terms of the geometry of the polyhedron P. As an ingredient in the computation, we explicitly calculate some of the lower K-groups of the dihedral groups and the product of dihedral groups with the cyclic group of order two.
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A hyperbolic 3-simplex reflection group is a Coxeter group arising as a lattice in the isometry group of hyperbolic 3-space, with fundamental domain a geodesic simplex (possibly with some ideal vertices). 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.
We show that the Waldhausen trace map $mathrm{Tr}_X colon A(X) to QX_+$, which defines a natural splitting map from the algebraic $K$-theory of spaces to stable homotopy, is natural up to emph{weak} homotopy with respect to transfer maps in algebraic $K$-theory and Becker-Gottlieb transfer maps respectively.
For $Gamma$ a relatively hyperbolic group, we construct a model for the universal space among $Gamma$-spaces with isotropy on the family VC of virtually cyclic subgroups of $Gamma$. We provide a recipe for identifying the maximal infinite virtually cyclic subgroups of Coxeter groups which are lattices in $O^+(n,1)= iso(mathbb H^n)$. We use the information we obtain to explicitly compute the lower algebraic K-theory of the Coxeter group $gt$ (a non-uniform lattice in $O^+(3,1)$). Part of this computation involves calculating certain Waldhausen Nil-groups for $mathbb Z[D_2]$, $mathbb Z[D_3]$.
Tate cohomology (as well as Borel homology and cohomology) of connective K-theory for $G=(mathbb{Z}/2)^n$ was completely calculated by Bruner and Greenlees. In this note, we essentially redo the calculation by a different, more elementary method, and we extend it to $p>2$ prime. We also identify the resulting spectra, which are products of Eilenberg-Mac Lane spectra, and finitely many finite Postnikov towers. For $p=2$, we also reconcile our answer completely with the result of Bruner and Greenlees, which is in a different form, and hence the comparison involves some non-trivial combinatorics.
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 C.
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