Related papers: Buildings have finite asymptotic dimension
We examine asymptotic dimension and property A for groups acting on complexes. In particular, we prove that the fundamental group of a finite, developable complex of groups will have finite asymptotic dimension provided the geometric…
We show for a free action of a countable group $\Gamma$ on a finite-dimensional, compact metric space by homeomorphisms that the dynamic asymptotic dimension is either infinite or coincides with the asymptotic dimension of $\Gamma$.
Let $X$ be a proper metric space and let $F$ be a finite group acting on $X$ by isometries. We show that the asymptotic dimension of $F\backslash X$ is the same as the asymptotic dimension of $X$.
A subset of the finite dimensional hypercube is said to be equilateral if the distance of any two distinct points equals a fixed value. The equilateral dimension of the hypercube is defined as the maximal size of its equilateral subsets. We…
We prove a version of the countable union theorem for asymptotic dimension and we apply it to groups acting on asymptotically finite dimensional metric spaces. As a consequence we obtain the following finite dimensionality theorems. A) An…
By recognizing them as fundamental groups of developable complexes of groups we prove that mapping class groups of compact orientable surfaces have finite asymptotic dimension.
In this paper we prove that the asymptotic dimension of a finite-dimensional CAT(0) cube complex is bounded above by the dimension. To achieve this we prove a controlled colouring theorem for the complex. We also show that every CAT(0) cube…
We show that the asymptotic dimension of a geodesic space that is homeomorphic to a subset in the plane is at most three. In particular, the asymptotic dimension of the plane and any planar graph is at most three.
Suppose that a finitely generated group $G$ is hyperbolic relative to a collection of subgroups $\{H_1, ..., H_m\} $. We prove that if each of the subgroups $H_1, ..., H_m$ has finite asymptotic dimension, then asymptotic dimension of $G$…
It is proven that if a finitely presented group is one ended it has asymptotic dimension bigger than one. It follows that finitely presented groups with asdim 1 are virtually free. A counterexample is given for the finitely generated case.
The purpose of this note is to characterize the asymptotic dimension $asdim(X)$ of metric spaces $X$ in terms similar to Property A of Yu: If $(X,d)$ is a metric space and $n\ge 0$, then the following conditions are equivalent: [a.]…
We prove equivalence of certain axiom sets for affine buildings. Along the lines a purely combinatorial proof of the existence of a spherical building at infinity is given. As a corollary we obtain that ``being an affine building'' is…
We construct a metric space whose transfinite asymptotic dimension and complementary-finite asymptotic dimension $2\omega+1$.
We show that the type function of a space with finite asymptotic dimension estimates its Hilbert (or any $l^p$) compression. The method allows to obtain the lower bound of the compression of the lamplighter group $Z\wr Z$, which has…
We construct a universal space for the class of proper metric spaces of bounded geometry and of given asymptotic dimension. As a consequence of this result, we establish coincidence of the asymptotic dimension with the asymptotic inductive…
In this paper we show that an instance of dividing in pseudofinite structures can be witnessed by a drop of the pseudofinite dimension. As an application of this result we give new proofs of known results for asymptotic classes of finite…
We introduce a quasi-symmetry invariant of a metric space Z called the capacity dimension. Our main result says that for a visual Gromov hyperbolic space X the asymptotic dimension of X is at most the capacity dimension of its boundary at…
We construct a class of metric spaces whose transfinite asymptotic dimension and complementary-finite asymptotic dimension are both $\omega+k$ for any $k\in\mathbb{N}$, where $\omega$ is the smallest infinite ordinal number and a metric…
We introduce the notion of asymptotic cohomology based on the bounded cohomology and define cohomological asymptotic dimension $\as_{\Z} X$ of metric spaces. We show that it agrees with the asymptotic dimension $\as X$ when the later is…
We investigate asymptotically flat manifolds with cone structure at infinity. We show that any such manifold M has a finite number of ends. For simply connected ends we classify all possible cones at infinity, except for the 4-dimensional…