Lower algebraicK-theory of certain reflection groups

Abstract

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.Comment: 35 pages, 2 figure

The Lower Algebraic K-Theory of Split Three-Dimensional Crystallographic Groups

“…where p is prime and s is the number of simple components A of QG with even Schur index but with A P of odd Schur index for each prime ideal P of the center of A that divides |G| (see [LMO10]). We first recall that the group algebra Q[Z/6] decomposes into simple components as follows:…”

The Whitehead group of (almost) extra-special p-groups with p odd

“…The computation of the Whitehead group W h(G) of a finite group G is in general very hard, and a compendium on the subject is the book by Bob Oliver ([17]) of 1988. Since then, it seems that not much progress has been made on this subject (see however [16], [15], [22], [21]).…”

The Lower Algebraic K-Theory of Virtually Cyclic Subgroups of the Braid Groups of the Sphere and of ZB4(S2)

“…This conjecture has been verified for a number of classes of groups, such as discrete cocompact subgroups of virtually connected Lie groups [33], finitely-generated Fuchsian groups [11], Bianchi groups [9], pure braid groups of aspherical surfaces [3], braid groups of aspherical surfaces [35] and for some classes of mapping class groups [10]. In [63], Lafont and Ortiz presented explicit computations of the lower algebraic K-theory of hyperbolic 3-simplex reflection groups, and then together with Magurn, for that of certain reflection groups [60]. Similar calculations were performed for virtually free groups in [54].…”

“…In Sections 2.3 and 2.5, we calculate the Whitehead and the K ´1-groups respectively of the integral group rings of many of the finite subgroups of B n pS 2 q. To our knowledge, these sections contain a number of original results, as well as some new phenomena, such as the existence of torsion for some K ´1-groups, that did not appear in previous work [54,60,63]. This necessitates alternative techniques, notably the application of results of Yamada to determine local Schur indices [84,85], which enables us to calculate the torsion of our K ´1-groups.…”

“…In the ensuing quest to compute the lower algebraic K-theory of the group ring ZrB n pS 2 qs, we encountered a number of difficulties, among them: (a) the family of virtually cyclic subgroups of B n pS 2 q is relatively large, and depends on n, contrasting sharply with the case of the pure braid groups analysed in [56]. (b) the lower algebraic K-theory of even the finite subgroups of B n pS 2 q is poorly understood, and the investigation of the K-groups of dicyclic and binary polyhedral groups presents additional technical obstacles compared to that of the dihedral and polyhedral groups that appear in [60,63] for example. (c) in order to apply the method of calculation suggested by the fibred isomorphism conjecture, one needs not only to compute the various Nil groups, but also to discover a suitable universal space for the family of virtually cyclic subgroups of B n pS 2 q.…”