The Hausdorff distance measures how far apart two sets are in a common metric space. By contrast, the Gromov-Hausdorff distance provides a notion of distance between two abstract metric spaces. How do these distances behave for quotients of spaces under group actions? Suppose a group G acts by isometries on two metric spaces X and Y. In this article, we study how the Hausdorff and Gromov-Hausdorff distances between X and Y and their quotient spaces X/G
Nearby in the stack
and
Y/G
are related. For the Hausdorff distance, we show that if
X
and
Y
are
G
-invariant subsets of a common metric space, then we have
dH(X,Y)=dH(X/G,Y/G)
. However, the Gromov-Hausdorff distance does not preserve this relationship: we show how to make the ratio
dGH(X,Y)dGH(X/G,Y/G)
both arbitrarily large and arbitrarily small, even if