Assouad-Nagata dimension of tree-graded spaces

2009/10/13 by Brodskiy, N., Higes, J.
#54F45 #FOS: Mathematics #Geometric Topology (math.GT) #Group Theory (math.GR) #Primary: 20F69 #Secondary: 54E35

paper · doi:10.48550/arxiv.0910.2378

Abstract

Given a metric space X of finite asymptotic dimension, we consider a quasi-isometric invariant of the space called dimension function. The space is said to have asymptotic Assouad-Nagata dimension less or equal n if there is a linear dimension function in this dimension. We prove that if X is a tree-graded space (as introduced by C. Drutu and M. Sapir) and for some positive integer n a function f serves as an n-dimensional dimension function for all pieces of X, then the function 300⋅ f serves as an n-dimensional dimension function for X. As a corollary we find a formula for the asymptotic Assouad-Nagata dimension of the free product of finitely generated infinite groups: asdim_AN (G*H)= max\asdim_AN (G), asdim_AN (H)\.

Related

• On Asymptotic Assouad-Nagata Dimension
• Nagata dimension, quasisymmetric embeddings, and Lipschitz extensions
• Asymptotic and Assouad-Nagata dimension of finitely generated groups and their subgroups
• Assouad-Nagata dimension of locally finite groups and asymptotic cones
• Assouad-Nagata dimension of wreath products of groups