1998/03/01 by Guoliang Yu · 1 citation
Mathematics · #Advanced Operator Algebra Research #Geometric and Algebraic Topology #Mathematical Dynamics and Fractals #Mathematics #Novikov self-consistency principle #Conjecture #Dimension (graph theory) #Pure mathematics
paper · doi:10.2307/121011
openalex publication_date 1998/03/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/22
Recall that the asymptotic dimension is a coarse geometric analogue of the covering dimension in topology (page 28, [14]). More precisely, the asymptotic dimension for a metric space is the smallest integer n such that for any r > 0, there exists a uniformly bounded cover C = UiiEI of the metric space for which the r-multiplicity of C is at most n + 1; i.e., no ball of radius r in the metric space intersects more than n + 1 members of C [14]. The class of finitely generated discrete groups with finite asymptotic dimension is hereditary in the sense that if a finitely generated group has finite asymptotic dimension as metric space with a word-length metric, then its finitely generated subgroups also have finite asymptotic dimension as metric spaces with word-length metrics (cf. Section 6). This, together with a result of Gromov in [14], implies that finitely generated subgroups of Gromov's hyperbolic groups have finite asymptotic dimension. Currently no example of a finitely generated group with infinite asymptotic dimension and finite classifying space is known. It should also be noted that two different definitions of asymptotic dimension