New algorithms for deciding the injectivit y of the global update funct ion associat ed with a cellular automaton C A of dimension one are presented. This is done by interpreting each ordered pair dete rmined by the local update function as th e edge of a labeled directed graph GR which has the prope rty that each bi-infinite sequence of states of the cellular automaton is th e sequence of inpu...

Recall that the rank of a finitely generated group is the minimal number of elements needed to generate it. M. White [Whi02] has proven that the injectivity radius of M is bounded above by some function of rank(π1(M)). Building on a technique that he introduced, we show that if M is a hyperbolic 3-manifold fibering over the circle with fiber Σg and rank(π1(M)) 6= 2g + 1, then the diameter of M ...

Let M be a closed simply connected manifold and 0 < δ ≤ 1. Klingenberg and Sakai conjectured that there exists a constant i0 = i0(M, δ) > 0 such that the injectivity radius of any Riemannian metric g on M with δ ≤ Kg ≤ 1 can be estimated from below by i0. We study this question by collapsing and Alexandrov space techniques. In particular we establish a bounded version of the KlingenbergSakai co...

This paper explores and ties together three themes. The first is to establish regularity of a metric tensor, on a manifold with boundary, on which there are given Ricci curvature bounds, on the manifold and its boundary, and a Lipschitz bound on the mean curvature of the boundary. The second is to establish geometric convergence of a (sub)sequence of manifolds with boundary with such geometrica...

Recall that a cover U is of dimension N if every x 2 X is contained in no more then N C 1 members of U . The asymptotic dimension of a finitely generated group is its asymptotic dimension as a metric space with respect to any word metric. An important result of Yu [19] asserts that the Novikov conjecture holds for groups of finite asymptotic dimension. This can be viewed as an injectivity resul...

We give a lower bound on the boundary injectivity radius of Margulis tubes with smooth constructed by Buser, Colbois, and Dodziuk. This estimate depends dimension curvature only.

let φ be a w  -continuous homomorphism from a dual banach algebra to c. the notion of φ-connes amenability is studied and some characterizations is given. a type of diagonal for dual banach algebras is dened. it is proved that the existence of such a diagonal is equivalent to φ-connes amenability. it is also shown that φ-connes amenability is equivalent to so-called φ-splitting of a certain s...

we survey the recent investigations on (bounded, sequential) approximate connesamenability and pseudo-connes amenability for dual banach algebras. we will discuss thecore problems concerning these notions and address the signicance of any solutions to themto the development of the eld.