  We introduce new obstructions to topological knot concordance. These are
obtained from amenable groups in Strebel's class, possibly with torsion, using
a recently suggested $L^2$-theoretic method due to Orr and the author.
Concerning $(h)$-solvable knots which are defined in terms of certain Whitney
towers of height $h$ in bounding 4-manifolds, we use the obstructions to reveal
new structure in the knot concordance group not detected by prior known
invariants: for any $n>1$ there are $(n)$-solvable knots which are not
$(n.5)$-solvable (and therefore not slice) but have vanishing
Cochran-Orr-Teichner $L^2$-signature obstructions as well as Levine algebraic
obstructions and Casson-Gordon invariants.
