Roundness of metric spaces was introduced by Per Enflo as a tool to study uniform structures of linear topological spaces. The present paper investigates geometric and topological properties detected by the roundness of general metric spaces. In particular, we show that geodesic spaces of roundness 2 are contractible, and that a compact Riemannian manifold with roundness >1 must be simply connected. We then focus our investigation on Cayley graphs of finitely generated groups. One of our main results is that every Cayley graph of a free Abelian group on 2 generators has roundness = 1. We show that if a group has no Cayley graph of roundness = 1, then it must be a torsion group with every element of order 2, 3, 5, or 7.
Mathematics Subject Classifications (2000). Primary: 20F65, Secondary: 57M07, 46B20.
The purpose of this article is to record the center of the Lie algebra obtained from the descending central series of Artin's pure braid group, a Lie algebra analyzed in work of Kohno [T. Kohno, Linear representations of braid groups and classical Yang-T. Kohno, Série de Poincaré-Koszul associée aux groupes de tresses pures, Invent. Math. 82 (1985) 57-75], and Falk and Randell [M. Falk, R. Randell, The lower central series of a fiber-type arrangement, Invent. Math. 82 (1985) 77-88]. The structure of this center gives a Lie algebraic criterion for testing whether a homomorphism out of the classical pure braid group is faithful which is analogous to a criterion used to test whether certain morphisms out of free groups are faithful [F.R. Cohen, J. Wu, On braid groups, free groups, and the loop space of the 2-sphere, in: Algebraic Topology: Categorical Decomposition Techniques, in: Progr. Math., vol. 215, Birkhäuser, Basel, 2003; Braid groups, free groups, and the loop space of the 2-sphere, math.AT/0409307]. However, it is as unclear whether this criterion for faithfulness can be applied to any open cases concerning representations of P n such as the Gassner representation.
Abstract.
We show that the holomorph of the free group on two generators satisfies the Farrell–Jones
Fibered Isomorphism Conjecture. As a consequence, we show that the lower K-theory of the above group vanishes.
We formulate and prove a geometric version of the Fundamental Theorem of Algebraic K-Theory which relates the K-theory of the Laurent polynomial extension of a ring to the K-theory of the ring. The geometric version relates the higher simple homotopy theory of the product of a nite complex and a circle with that of the complex. By using methods of controlled topology, we also obtain a geometric version of the Fundamental Theorem of Lower Algebraic K-Theory. The main new innovationis a geometricallyde ned Nil space. 2 BRUCE HUGHES AND STRATOS PRASSIDIS 6.11. Bounded Whitehead and Pseudoisotopy Spaces 7. Torsion and a higher sum theorem 8. Nil as a geometrically de ned simplicial set 8.1. Preliminaries 8.2. The simplicial set of nil simplices 8.3. The simplicial maps p :
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.