Rough isometries

An isometry is a map {f\colon X\rightarrow Y} between two metric spaces {X,Y} which preserves all distances: {d_Y(f(x),f(x')) = d_X(x,x')} for all {x,x'\in X}. (After typing a bunch of such formulas, one tends to prefer shorter notation: {|f(x)f(x')| = |xx'|}, with the metric inferred from contexts.) It’s a popular exercise to prove that every isometry from a compact metric space into itself is surjective.

A rough isometry allows additive distortion of distances: {|\,|f(x)f(x')| - |xx'|\,|\le \epsilon}. (As contrasted with bi-Lipschitz maps, which allow multiplicative distortion). Rough isometries ignore small-scale features (in particular, they need not be continuous), but accurately capture the large scale structure of a space. This makes them convenient for studying hyperbolic spaces (trees and their relatives), where the large-scale structure is of interest.

If {f} is an {\epsilon}-rough isometry, then the Gromov-Hausdorff distance {d_{GH}} between {X} and {f(X)} is at most {\epsilon/2}. This follows from a convenient characterization of {d_{GH}}: it is equal to {\frac12 \inf_R \textrm{dis}\,(R)} where the infimum is taken over all subsets {R\subset X\times Y} that project surjectively onto each factor, and {\textrm{dis}\,(R) = \sup \{|\,|xx'|-|yy'|\,| \colon xRy, \ x'Ry' \} }.

Since small-scale features are ignored, it is not quite natural to talk about rough isometries being surjective. A natural concept for them is {\epsilon}-surjectivity, meaning that every point of {Y} is within distance {\le \epsilon} of {f(X)}. When this happens, we can conclude that {d_{GH}(X,Y)\le 3\epsilon/2}, because the Hausdorff distance between {Y} and {f(X)} is at most {\epsilon}.

Recalling that an isometry of a compact metric space {X} into {X} is automatically onto, it is natural to ask whether {\epsilon}-rough isometries of {X} into {X} are {\epsilon}-surjective. This, however, turns out to be false in general.

First example, a finite space: {X=\{0,1,2,\dots,n\}} with the metric {d(i,j) = i+j} (when {i\ne j}). Consider the backward shift {f(i)=i-1} (and {f(0)=0}). By construction, this is a rough isometry with {\epsilon=2}. Yet, the point {n} is within distance {n} from {f(X) = \{0,1,2,\dots, n-1\}}.

The above metric can be thought of a the distance one has to travel from {i} to {j} with a mandatory visit to {0}. This makes it similar to the second example.

Second example, a geodesic space: {X} is the graph shown below, a subset of {\mathbb R^2} with the intrinsic (path) metric, i.e., the length of shortest path within the set.

Comb space
Comb space

Define {f(x,y) = ((x-1)^+, (y-1)^+)} where {{\,}^+} means the positive part. Again, this is a {2}-rough isometry. The omitted part, shown in red below, contains a ball of radius {5} centered at {(5,5)}. Of course, {5} can be replaced by any positive integer.

Omitted part in red
Omitted part in red

Both of these examples are intrinsically high-dimensional: they cannot be accurately embedded into {\mathbb R^d} for low {d} (using the restriction metric rather than the path metric). This raises a question: does there exist {C=C(d)} such that for every compact subset {K\subset \mathbb R^d}, equipped with the restriction metric, every {\epsilon}-rough isometry {f\colon K\rightarrow K} is {C(d)\epsilon}-surjective?