On asymptotic dimension of groups

By Gregory C. Bell, Alexander Dranishnikov. 2000

105 citations recorded by the source corpus.

We prove a version of the countable union theorem for asymp- totic dimension and we apply it to groups acting on asymptotically nite dimensional metric spaces. As a consequence we obtain the following nite dimensionality theorems. A) An amalgamated product of asymptotically nite dimensional groups has nite asymptotic dimension: asdimAC B< 1 . B) Suppose that G 0 is an HNN extension of a group G with asdimG < 1. Then asdimG 0 <1. C) Suppose that is Davis' group constructed from a group with asdim <1. Then asdim <1. AMS Classication 20H15; 20E34, 20F69

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/91fd2f1c42b82b14fc7518ebf87d9d9d63048d9b

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Citations in the corpus

Dimension of asymptotic cones of Lie groups (2007, background): “Other kinds of dimensions that provide quasi-isometry invariants are, among others, asymptotic dimension [Gro, BeDr ], subexponential corank [BuSc1], hyperbolic dimension [BuSc2].”
Dimensions of locally and asymptotically self-similar spaces (2005, methodology): “The following lemma implies in particular the finite union theorem for the capacity dimension, which is similar to the appropriate theorems for the asymptotic dimension [ BD ] and the Assouad-Nagata dimension [LS].”
On Asymptotic Dimension of Groups Acting on Trees (2001, methodology): “We will use the following theorems proven in [ BD ].”
Metric sparsification and operator norm localization (2007, methodology): “We follow the strategy in Bell and Dranishnikov [2].”
The integral K-theoretic Novikov conjecture for groups with finite asymptotic dimension (2004, background): “Examples from this apparently very large class are the Gromov hyperbolic groups [9], Coxeter groups [8], various generalized products of these, including the groups acting on trees with vertex stabilizers of finite asymptotic dimension [2], and, more generally, fundamental groups of developable comp”
Uniform embeddability of relatively hyperbolic groups (2005, methodology): “This approach to permanence results originated with the work of Bell and Dranishnikov [2], [3].”
Weak hyperbolicity of cube complexes and quasi‐arboreal groups (2011, background): “ dimension of cube complexes. In this section, we discuss the asymptotic dimension of groups acting on CAT(0) cube complexes and relate this to quasi-arboreality. Deﬁnition 6.1 (Asymptotic dimension, [BD01]). Let (M,d) be a metric space. The asymptotic dimension of M is at most nif for each r>0 t”
On homological coherence of discrete groups (2004, background): “Exam from this apparently very large class are the Gromov hyperbolic groups [10], Co groups [9], various generalized products of these, including the groups acting on tree vertex stabilizers of finite asymptotic dimension [2], and, more generally, fundam groups of developable complexes of finite-dim”
Universal spaces for asymptotic dimension (2002, background): “[2] Assume that X = ∪αXα and asdimXα uniformly.”
A notion of geometric complexity and its application to topological rigidity (2010, background): “ity is the smallest collection containing bounded metric families, and stable under weak decomposition. Clearly, both nite asymptotic dimension (again, uniformly in the sense of Bell and Dranishnikov [BD1]) and nite decomposition complexity imply weak nite decomposition complexity. While true that n”
Squeezing and Higher Algebraic K-Theory (2003, background): “For more information on groups of finite asymptotic dimension see for example [DJ99] and [BD01].”
A Hurewicz-type theorem for asymptotic dimension and applications to geometric group theory (2004, methodology): “What follows is a natural generalization of the combinatorial structure of amalgamated free products studied in [ 2 ].”
Asymptotic dimension of relatively hyperbolic groups (2004, background): “The lemma below appears as Theorem 2 in [1].”
Groups acting on trees with almost prescribed local action (2015, unknown): “ro [CH15, Proposition 4.D.4], and therefore by the previous observation Wr(x0) (endowed with the induced topology) has asymptotic dimension zero as well. So we are in position to apply Theorem 2 from [BD01], which implies that Ghas asymptotic dimension at most one. Note that the result is stated the”