Most of the
years of study of the set of knot concordance classes,
, has
focused on its structure as an abelian group. Here we take a different approach, namely
we study
as a metric space admitting many natural geometric operators. We focus
especially on the coarse geometry of satellite operators. We consider
several knot concordance spaces, corresponding to different categories of
concordance, and two different metrics. We establish the existence of
quasi-–flats
for every
,
implying that
admits no quasi-isometric embedding into a finite product of (Gromov) hyperbolic
spaces. We show that every satellite operator is a quasihomomorphism
. We
show that winding number one satellite operators induce quasi-isometries with
respect to the metric induced by slice genus. We prove that strong winding
number one patterns induce isometric embeddings for certain metrics. By
contrast, winding number zero satellite operators are bounded functions and
hence quasicontractions. These results contribute to the suggestion that
is a
fractal space. We establish various other results about the large-scale geometry of
arbitrary satellite operators.
