Abstract: Given a metric space $X$ of finite asymptotic dimension, we consider aquasi-isometric invariant of the space called dimension function. The space issaid to have asymptotic Assouad-Nagata dimension less or equal $n$ if there isa linear dimension function in this dimension. We prove that if $X$ is atree-graded space as introduced by C. Drutu and M. Sapir and for somepositive integer $n$ a function $f$ serves as an $n$-dimensional dimensionfunction for all pieces of $X$, then the function $300\cdot f$ serves as an$n$-dimensional dimension function for $X$. As a corollary we find a formulafor the asymptotic Assouad-Nagata dimension of the free product of finitelygenerated infinite groups: $asdim {AN} G*H= max\{asdim {AN} G, asdim {AN}H\}.$

Author: N. Brodskiy, J. Higes

Source: https://arxiv.org/