Universal spaces for asymptotic dimension (Q1882923)

Let us begin by recalling the definition of asymptotic dimension for a metric space \(X\), a concept that was introduced by \textit{M. Gromov} [Geometric group theory. Volume 2: Asymptotic invariants of infinite groups, Lond. Math. Soc. Lecture Notes 182 (1993; Zbl 0841.20039)]. Given \(n\geq 0\), one denotes asdim\(X\leq n\) if for every \(D>0\), there exists a uniformly bounded cover \(\mathcal U\) of \(X\) such that \(\mathcal U=\mathcal U^0\cup\dots\cup\mathcal U^n\) and for each \(0\leq i\leq n\), \(\mathcal U^i\) is \(D\)-disjoint. Here \(\mathcal U\) being uniformly bounded means that for some \(C>0\), diam\((U)<C\) for all \(U\in\mathcal U\), and \(D\)-disjoint that there exists \(\varepsilon>0\) such that for each \(U\), \(V\in\mathcal U^i\) with \(U\neq V\), \(x\in U\), and \(y\in V\), \(d(x,y)\geq D+\varepsilon\). Asymptotic dimension is important in the study of finitely presented groups considered as metric spaces with the word metric. In section 5 of the paper, the authors recall the notion of large asymptotic inductive dimension, asInd from [\textit{A. Dranishnikov}, JP J. Geom. Topol. 1, 239--247 (2001; Zbl 1059.54024)]. Roughly speaking, its definition runs parallel to that of large inductive dimension, Ind, but we shall not state it here. There is similarly small asymptotic inductive dimension, asind, in parallel with small inductive dimension, ind. A metric space \((X,d)\) is called proper if for each \(x\in X\), the map of \(X\) to the reals sending each \(y\in X\) to \(d(y,x)\) is a proper map. A subset \(A\) is called \(r\)-discrete \((r>0)\) if for each two points \(x\), \(y\in A\), \(d(x,y)\geq r\); the \(r\)-capacity of \(A\) is the maximum cardinality of an \(r\)-discrete subset of \(A\). One then says that \((X,d)\) is of bounded geometry if there is \(r>0\) and a function \(c:[0,\infty)\rightarrow[0,\infty)\) such that for every \(x\in X\), the \(\varepsilon\)-ball centered at \(x\) has \(r\)-capacity at most \(c(\varepsilon)\). The authors prove, Theorem 3.10, that for every \(n\), there is a separable metric space \(M_n\), whose asymptotic dimension is \(n\), and which is universal for proper metric spaces \(X\) with bounded geometry and asdim\(X\leq n\). One caveat here is that the spaces \(M_n\) themselves are neither proper nor of bounded geometry. Using this result, they demonstrate (see section 5) that in the class of proper metric spaces with finite asymptotic dimension, all three dimension functions asdim, asInd, and asind coincide.