The Farrell-Hsiang method revisited

We present a sufficient condition for groups to satisfy the Farrell-Jones Conjecture in algebraic K-theory and L-theory. The condition is formulated in terms of finite quotients of the group in question and is motivated by work of Farrell-Hsiang.Comment: This version is different from the published version. A number of typos and an incorrect formula for the transfer before Lemma 6.3 pointed out by Holger Reich have been correcte

On Universal Tilers

A famous problem in discrete geometry is to find all monohedral plane tilers, which is still open to the best of our knowledge. This paper concerns with one of its variants that to determine all convex polyhedra whose every cross-section tiles the plane. We call such polyhedra universal tilers. We obtain that a convex polyhedron is a universal tiler only if it is a tetrahedron or a pentahedron.Comment: 10 pages, 2 figure

Multi-triangulations as complexes of star polygons

Maximal $(k+1)$-crossing-free graphs on a planar point set in convex position, that is, $k$-triangulations, have received attention in recent literature, with motivation coming from several interpretations of them. We introduce a new way of looking at $k$-triangulations, namely as complexes of star polygons. With this tool we give new, direct, proofs of the fundamental properties of $k$-triangulations, as well as some new results. This interpretation also opens-up new avenues of research, that we briefly explore in the last section.Comment: 40 pages, 24 figures; added references, update Section

Optimally Dense Packings for Fully Asymptotic Coxeter Tilings by Horoballs of Different Types

The goal of this paper to determine the optimal horoball packing arrangements and their densities for all four fully asymptotic Coxeter tilings (Coxeter honeycombs) in hyperbolic 3-space $\mathbb{H}^3$. Centers of horoballs are required to lie at vertices of the regular polyhedral cells constituting the tiling. We allow horoballs of different types at the various vertices. Our results are derived through a generalization of the projective methodology for hyperbolic spaces. The main result states that the known B\"or\"oczky--Florian density upper bound for "congruent horoball" packings of $\mathbb{H}^3$ remains valid for the class of fully asymptotic Coxeter tilings, even if packing conditions are relaxed by allowing for horoballs of different types under prescribed symmetry groups. The consequences of this remarkable result are discussed for various Coxeter tilings.Comment: 26 pages, 10 figure

Algorithmic Aspects of tree amalgamation

The amalgamation of leaf-labelled trees into a single (super)tree that "displays" each of the input trees is an important problem in classification. We discuss various approaches to this problem and show that a simple and well known polynomial-time algorithm can be used to solve this problem whenever the input set of trees contains a minimum size subset that uniquely determines the supertree. Our results exploit a recently established combinatorial property concerning the structure of such collections of trees

Zum Problem der regelmaessigen Flaechenaufteilung Anmerkungen zu M.C. Eschers Aufsatz 'Regelmatige vlakverdeling' aus der Sicht eines Mathematikers

Copy held by FIZ Karlsruhe; available from UB/TIB Hannover / FIZ - Fachinformationszzentrum Karlsruhe / TIB - Technische InformationsbibliothekSIGLEDEGerman