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Let \((X_{n})_{n \in \mathbb{N}}\) be a sequence of finite graphs with
corresponding standard graph metrics \((d_{n})_{n \in \mathbb{N}}\) and
\((X, d)\) be the coarse disjoint union of \(\{ (X_{n}, d_{n})\}_{n \in
\mathbb{N}}\).
There exists a set of cut metrics \((d_{n,w})_{n \in \mathbb{N}}\) on
\((X_{n})_{n \in \mathbb{N}}\) such that the identity function from
\((X,d)\) to the coarse disjoint union \((X, d_{w})\) of \(\{(X_{n},
d_{n,w})\}_{n \in \mathbb{N}}\) is a coarse embedding if and only if
there exists a coarse disjoint union, \((C,d_{C})\), of a sequence of
hypercubes, \((C_{n})_{n \in \mathbb{N}}\), each equipped with the
standard graph metric, \((d_{n,C})_{n \in \mathbb{N}}\), and a coarse
embedding function \(f:(X, d) \to (C,d_{C})\).
Assume there exists a cut metric \(d_{w}\) on \(X\) such that \(id:(X,d)
\to (X,d_{w})\) is a coarse embedding. Since each \((X_{n}, d_{n,w})\)
isometrically embeds into a hypercube equipped with the standard
graph metric, \((C_{n},
d_{n,C})\), \((X,d)\) coarsely
embeds into the disjoint union of \(\{(C_{n}, d_{n, C})\}_{n \in
\mathbb{N}}\)~\cite[Proposition~4.2.2]{deza:geometry}.
Conversely, assume there exists a coarse disjoint union, \((C,
d_{C})\), of a sequence of hypercubes, \((C_{n}, d_{n,C})_{n \in
\mathbb{N}}\), equipped with the standard graph metric and a coarse
embedding function \(f:(X, d) \to (C,d_{C})\) such that \(f(X_{n})
\subseteq C_{n}\) for all \(n \in \mathbb{N}\). Using the Hamming
encoding of a hypercube we can create partitions of the vertices of
each \(C_{n}\) such that for any \(x,y \in X_{n}\), \(d_{n,w}(x,y) =
d_{n,C}(f(x),f(y))\).
Fix an \(n \in \mathbb{N}\). \(C_{n}\) contains \(2^m\) vertices for some
\(m \in \mathbb{N}\). Define partitions \(S^n_{1},\ldots,S^n_{m}\) of
the vertices of \(X_{n}\) by
\begin{equation*}
S^n_{k} := \{ x : x \in X_{n}, h(f(x))_{k} = 1 \}.
\end{equation*}
The set of partitions \(S^n_{1},\ldots,S^n_{m}\) defines a cut metric,
\(d_{n,w}\), on \(X_{n}\) by
\begin{equation*}
d_{n,w}(x,y) = |\{ k : |S^n_{k} \cap \{ x,y \}| = 1 \}|, ~\forall x,y \in X_{n}
\end{equation*}
which is equal to the number of bits which differ between
\(h(f(x))\) and \(h(f(y))\). Hence, for all \(n \in \mathbb{N}\) and
all \(x,y \in X_{n}\), \(d_{n,w}(x,y) = d_{n,C}(f(x), f(y))\).
Let \(d_{w}:X \times X \to \mathbb{R}^{\geq 0}\) be defined such
that \(d_{w}\restrict{X_{n}} = d_{n,w}\) and \(d_{w}(X_{n}, X_{m}) =
d_{C}(f(X_{n}), f(X_{m}))\) for all \(m,n \in \mathbb{N}\). Then, for any
\(x,y \in X\), \(d_{w}(x,y) = d_{C}(f(x),f(y))\) hence \(id:(X, d)
\to (X, d_{w})\) is a coarse equivalence.